Random orthonormal bases of spaces of high dimension

TL;DR

This paper proves that random orthonormal bases of high-dimensional Hilbert spaces tend to a unique limit state, extending quantum ergodicity results to various geometric settings using Schur polynomial techniques.

Contribution

It generalizes quantum ergodicity to high-dimensional spaces and introduces a novel approach using moments of inertia and Schur polynomials.

Findings

Random ONBs tend to a unique limit state as dimension grows

Results apply to eigenspaces of Laplace or Schrödinger operators in high dimensions

Method uses moments of inertia and Schur polynomial calculations

Abstract

We consider a sequence H_N of Hilbert spaces of dimensions d_N tending to infinity. The motivating examples are eigenspaces or quasi-mode spaces of a Laplace or Schrodinger operator. We define a random ONB of H_N by fixing one ONB and changing it by a random element of U(d_N). A random ONB of the direct sum of the H_N is an independent sequence {U_N} of random ONB's of the H_N. We prove that if d_N tends to infinity and if the normalized traces of observables in H_N tend to a unique limit state, then a random ONB also tends to that limit state. This generalizes an earlier result of the author for eigenspaces of the standard 2-sphere, and shows that the result does not depend on how fast the dimensions grow. In particular it is valid for eigenspaces of a flat rational torus in dimensions > 4. The main idea is to convert quantum ergodicity into a problem on the moments of inertia of permutahedra and to calculate the moments using Schur polynomials.