Quantum ergodicity of random orthonormal bases of spaces of high dimension

Abstract

We consider a sequence of finite-dimensional Hilbert spaces of dimensions . Motivating examples are eigenspaces, or spaces of quasi-modes, for a Laplace or Schrödinger operator on a compact Riemannian manifold. The set of Hermitian orthonormal bases of may be identified with U ( d N ), and a random orthonormal basis of is a choice of a random sequence U N ∈ U ( d N ) from the product of normalized Haar measures. We prove that if and if tends to a unique limit state ω ( A ), then almost surely an orthonormal basis is quantum ergodic with limit state ω ( A ). This generalizes an earlier result of the author in the case where is the space of spherical harmonics on S 2 . In particular, it holds on the flat torus if d ≥5 and shows that a highly localized orthonormal basis can be synthesized from quantum ergodic ones and vice versa in relatively small dimensions.

Quantum unique ergodicity for random bases of spectral projections

“…The assumptions we place on the manifold are the same as those used in [11,12]. We first recall that the eigenvalues of P have a predictable asymptotic distribution, so we can control the dimension of the spaces H k .…”

“…Towards this end, several models for a random sequence of functions have been proposed to simulate various limiting behaviors. In a series of articles, Zelditch [10,11,12] introduced a random matrix model to analyze the limiting behavior of a random sequence of functions which are short linear combinations of eigenfunctions, each with eigenvalue λ j growing to infinity. Let us now recall this model.…”

“…Now we can recall the previous results of Zelditch on this random wave model. Theorem 1.1 (Zelditch [10,11,12]). Suppose the spectral projections Π k satisfy dim H k → ∞ and the Weyl asymptotics…”

Quantum ergodicity of Wigner induced random spherical harmonics

“…This fact notwithstanding, it is shown in [Z2] that a random orthonormal basis (defined using Haar measures on unitary groups) of spherical harmonics is almost surely quantum ergodic, a result that is extended to Laplacian eigenfunctions on compact Riemannian manifolds in [Z3,Z4,M,BL]. The purpose of this paper is to return to the sphere and prove quantum ergodicity for a wider class of 'random' spherical harmonics.…”

Complex patterns in wave functions: drums, graphs and disorder

“…Some of the corresponding discussions are reflected in the remaining four original research contributions to this issue: Zelditch [9] discusses random orthonormal bases in large dimensional Hilbert spaces and proves that they are almost surely quantum ergodic, Oren & Smilansky [10] investigate spectral statistics of permutation matrices, Atas & Bogomolny [11] analyse the multi-fractal properties of wave functions in spin chains and Fyodorov & Keating [12] give a very detailed account of their asymptotic analysis of extreme value statistics for characteristic functions of random matrices and the Riemann zeta function.…”