Pär Kurlberg

Hecke theory and equidistribution for the quantization of linear maps of the torus

1.1. Background. One of the key issues of “Quantum Chaos” is the nature of the semiclassical limit of eigenstates of classically chaotic systems. When the classical system is given by the geodesic flow on a compact Riemannian manifoldM (or rather, on its cotangent bundle), one can formulate the problem as follows: The quantum Hamiltonian is, in suitable units, represented by the positive Laplacian − on M . To measure the distribution of its eigenstates, we start with a (smooth) classical observable, that is, a (smooth) function on the unit cotangent bundle S∗M; via some choice of quantization from symbols to pseudodifferential operators, we form its quantization Op(f ). This is a zero-order pseudodifferential operator with principal symbol f . The expectation value of Op(f ) in the eigenstate ψ is 〈Op(f )ψ,ψ〉. Letψj be a sequence of normalized eigenfunctions: ψj+λjψj = 0, ∫ M |ψj |2= 1. The problem then is to understand the possible limits as λj →∞ of the distributions f ∈ C∞(S∗M) −→ 〈Op(f )ψj ,ψj 〉. (1.1)

A local Riemann hypothesis, I

Daniel Bump1, Kwok-Kwong Choi2, Par Kurlberg3, Jeffrey Vaaler4 1 Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA (e-mail: bump@math.stanford.edu) 2 Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, Hong Kong (e-mail: choi@maths.hku.hk) 3 Department of Mathematics, University of Georgia, Athens, GA 30602, USA (e-mail: kurlberg@math.uga.edu) 4 University of Texas at Austin, Department of Mathematics RLM 8.100, Austin, TX 78712-1082, USA (e-mail: vaaler@math.utexas.edu)

The distribution of spacings between quadratic residues

We study the distribution of spacings between squares modulo q, where q is square-free and highly composite, in the limit as the number of prime factors of q goes to infinity. We show that all correlation functions are Poissonian, which among other things, implies that the spacings between nearest neighbors, normalized to have unit mean, have an exponential distribution. Date: Dec 14, 1998. Supported in part by a grant from the Israel Science Foundation. In addition, the first author was partially supported by the EC TMR network ”Algebraic Lie Representations”, EC-contract no ERB FMRX-CT97-0100. 1

Lattice points on circles and discrete velocity models for the Boltzmann equation

The construction of discrete velocity models or numerical methods for the Boltzmann equation, may lead to the necessity of computing the collision operator as a sum over lattice points. The collision operator involves an integral over a sphere, which corresponds to the conservation of energy and momentum. In dimension two there are difficulties even in proving the convergence of such an approximation since many circles contain very few lattice points, and some circles contain many badly distributed lattice points. However, by showing that lattice points on most circles are equidistributed we find that the collision operator can indeed be approximated as a sum over lattice points in the two-dimensional case. The proof uses a weak form of the Halberstam-Richert inequality for multiplicative functions (a proof is given in the paper), and estimates for the angular distribution of Gaussian primes. For higher dimensions, this result has already been obtained by Palczewski, Schneider, and Bobylev [SIAM J. Numer. Anal., 34 (1997), pp. 1865-1883].

Value distribution for eigenfunctions of desymmetrized quantum maps

We study the value distribution and extreme values of eigenfunctions for the “quantized cat map.” This is the quantization of a hyperbolic linear map of the torus. In a previous paper it was observed that there are quantum symmetries of the quantum map—a commutative group of unitary operators that commute with the map,which we called “Hecke operators.” The eigenspaces of the quantum map thus admit an orthonormal basis consisting of eigenfunctions of all the Hecke operators, which we call “Hecke eigenfunctions.”In this note we investigate suprema and value distribution of the Hecke eigenfunctions. For prime values of the inverse Planck constant N for which the map is diagonalizable modulo N (the “split primes” for the map), we show that the Hecke eigenfunctions are uniformly bounded and their absolute values (amplitudes) are either constant or have a semi-circle value distribution as N tends to infinity. Moreover, in the latter case different eigenfunctions become statistically independent. We obtain these results via the Riemann hypothesis for curves over a finite field (Weils theorem) and recent results of N. Katz on exponential sums. For general N we obtain a nontrivial bound on the supremum norm of these Hecke eigenfunctions.

On the order of unimodular matrices modulo integers

Assuming the Generalized Riemann Hypothesis, we prove the following: If b is an integer greater than one, then the multiplicative order of b modulo N is larger than N^(1-\epsilon) for all N in a density one subset of the integers. If A is a hyperbolic unimodular matrix with integer coefficients, then the order of A modulo p is greater than p^(1-\epsilon) for all p in a density one subset of the primes. Moreover, the order of A modulo N is greater than N^(1-\epsilon) for all N in a density one subset of the integers.

A gap principle for dynamics

Let f(1), ... , f(g) is an element of C(z) be rational functions, let Phi = (f(1), ... ,f(g)) denote their coordinate-wise action on (P-1)(g), let V subset of (P-1)(g) be a proper subvariety, and let P be a point in (P-1)(g)(C). We show that if S = {n >= 0 : Phi(n)(P) is an element of V(C)} does not contain any infinite arithmetic progressions, then S must be a very sparse set of integers. In particular, for any k and any sufficiently large N, the number of n <= N such that Phi(n)(P) is an element of V(C) is less than log(k)N, where log(k) denotes the kth iterate of the log function. This result can be interpreted as an analogue of the gap principle of Davenport-koth and Mumford.

Matrix elements for the quantum cat map: fluctuations in short windows

We study fluctuations of the matrix coefficients for the quantized cat map. We consider the sum of matrix coefficients corresponding to eigenstates whose eigenphases lie in a randomly chosen window, assuming that the length of the window shrinks with Plancks constant. We show that if the length of the window is smaller than the square root of Plancks constant, but larger than the separation between distinct eigenphases, then the variance of this sum is proportional to the length of the window, with a proportionality constant which coincides with the variance of the individual matrix elements corresponding to Hecke eigenfunctions.

On probability measures arising from lattice points on circles

A circle, centered at the origin and with radius chosen so that it has non-empty intersection with the integer lattice