Arbitrarily small perturbations of Dirichlet Laplacians are quantum unique ergodic

Abstract

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