Arbitrarily small perturbations of Dirichlet Laplacians are quantum unique ergodic

TL;DR

This paper demonstrates that tiny perturbations of the Dirichlet Laplacian on certain Euclidean domains can induce quantum unique ergodicity of high energy eigenfunctions, even with minimal boundary regularity.

Contribution

It introduces a method to achieve quantum unique ergodicity through arbitrarily small perturbations of the Laplacian, extending to domains with rough boundaries.

Findings

Existence of perturbations leading to QUE eigenfunctions.

High energy eigenfunctions become QUE under small perturbations.

Results hold for domains with minimal boundary regularity.

Abstract

Given an Euclidean domain with very mild regularity properties, we prove that there exist perturbations of the Dirichlet Laplacian of the form $-(I+S_\epsilon)\Delta$ with $\|S_\epsilon\|_{L^2\to L^2}\leq \epsilon$ whose high energy eigenfunctions are quantum uniquely ergodic (QUE). Moreover, if we impose stronger regularity on the domain, the same result holds with $\|S_\epsilon\|_{L^2\to H^\gamma}\leq \epsilon$ for $\gamma>0$ depending on the domain. We also give a proof of a local Weyl law for domains with rough boundaries.