Probabilistic Weyl laws for quantized tori

TL;DR

This paper proves a Weyl law for the expected eigenvalue count of small random perturbations in Toeplitz quantization on tori, with numerical evidence supporting its validity for pseudospectral eigenvalues.

Contribution

It establishes a probabilistic Weyl law for eigenvalues of perturbed quantized observables on tori, extending classical spectral asymptotics to a quantum setting.

Findings

Expected eigenvalue count follows a Weyl law.

Numerical experiments confirm the law for pseudospectral eigenvalues.

Results connect spectral theory with random perturbations in quantization.

Abstract

For the Toeplitz quantization of complex-valued functions on a $2n$-dimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural Weyl law. In numerical experiments the same Weyl law also holds for ``false'' eigenvalues created by pseudospectral effects.