Almost all eigenfunctions of a rational polygon are uniformly distributed

TL;DR

This paper proves that for a rational polygon, most eigenfunctions of the Dirichlet Laplacian become uniformly distributed in the domain as their eigenvalues grow large.

Contribution

It establishes that almost all eigenfunctions in a rational polygon have probability measures converging to uniform distribution, confirming quantum ergodicity in this setting.

Findings

Existence of a density-one subsequence of eigenfunctions converging to Lebesgue measure

Almost all eigenfunctions are uniformly distributed in the domain

Supports quantum ergodicity conjecture for rational polygons

Abstract

We consider an orthonormal basis of eigenfunctions of the Dirichlet Laplacian for a rational polygon. The modulus squared of the eigenfunctions defines a sequence of probability measures. We prove that this sequence contains a density-one subsequence that converges to Lebesgue measure.