Local Weak Limits of Laplace Eigenfunctions

TL;DR

This paper introduces local weak limits for Laplace eigenfunctions, providing a rigorous framework to analyze their asymptotic behavior and supporting Berry's random wave conjecture with new results.

Contribution

It defines local weak convergence for eigenfunctions, proves its existence on arbitrary domains, and connects it to the study of nodal domains and Berry's conjecture.

Findings

Deterministic eigenfunctions on a2^2 satisfy the random wave conjecture conclusions.

Any sequence of eigenfunctions admits local weak limits.

Local weak limits are useful for studying nodal domain asymptotics.

Abstract

In this paper, we introduce a new notion of convergence for the Laplace eigenfunctions in the semiclassical limit, the local weak convergence. This allows us to give a rigorous statement of Berry's random wave conjecture. Using recent results of Bourgain, Buckley and Wigman, we will prove that some deterministic families of eigenfunctions on $\mathbb{T}^2$ satisfy the conclusions of the random wave conjecture. We also show that on an arbitrary domain, a sequence of Laplace eigenfunctions always admits local weak limits. We explain why these local weak limits can be a powerful tool to study the asymptotic number of nodal domains.