Local Weak Limits of Laplace Eigenfunctions

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.

arXiv comment23 pages

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Cited by in corpus (4)

• Eigenfunctions and Random Waves in the Benjamini-Schramm limit
• Functional Convergence of Berry's Nodal Lengths: Approximate Tightness and Total Disorder
• Random moments for the new eigenfunctions of point scatterers on rectangular flat tori
• Eigenvectors of the square grid plus GUE