Eigenvalue statistics and lattice points

One of the more challenging problems in spectral theory and mathematical physics today is to understand the statistical distribution of eigenvalues of the Laplacian on a compact manifold. There are now several challenging conjectures about these, originating in the physics literature. In this survey, a version of a talk delivered as the Colloquio De Giorgi at the Scuola Normale Superiore of Pisa in May 2006, I will describe what is conjectured and what is known the very simple case of the flat torus, where the problems amount to counting lattice points in annuli and have a definite arithmetic flavour. Eigenvalue problems. Let M be a smooth, compact Riemannian manifold, the Laplace-Beltrami operator associated with the metric. We consider the eigenvalue problem ψ + λψ = 0, ψ ∈ L(M). As is well known, the spectrum is a discrete set of points, which we denote by 0 = λ0 ≤ λ1 ≤ λ2 ≤ . . . , whose only accumulation point is at infinity, and there is an orthonormal basis of L2(M) consisting of eigenfunctions. Example 1.1. The circle M = R/Z with the standard flat metric: the Laplacian is simply d2/dx2, as a basis for the eigenfunctions we may take sin 2πkx , cos 2πkx (k > 0) with eigenvalues 4π2k2 (each having multiplicity 2), together with the constant function 1 (eigenvalue zero). Example 1.2. The flat torus R2/Z2: here the Laplacian is = ∂2 ∂x2 + ∂2 ∂y2 and as an orthonormal basis of eigenfunctions we may take the exponentials exp(2π ik · x), k ∈ Z2, with eigenvalue 4π2|k|2. Spectral counting functions. We define the following spectral functions: A cumulative count n(x) := #{m : √λm ≤ x} Supported by a grant from the Israel Science Foundation.