Vol. 3, No. 3, 2021

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Local weak limits of Laplace eigenfunctions

Maxime Ingremeau

Vol. 3 (2021), No. 3, 481–515
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 𝕋2 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.

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Keywords
semiclassical analysis, quantum chaos, Berry's random waves conjecture, nodal domains, Benjamini–Schramm convergence
Mathematical Subject Classification 2010
Primary: 35P20
Milestones
Received: 2 February 2020
Revised: 9 July 2020
Accepted: 9 August 2020
Published: 13 May 2021
Authors
Maxime Ingremeau
Laboratoire J.A. Dieudonné UMR CNRS
Université Côte d’Azur
Nice
France