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2021 Local weak limits of Laplace eigenfunctions
Maxime Ingremeau
Tunisian J. Math. 3(3): 481-515 (2021). DOI: 10.2140/tunis.2021.3.481

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.

Citation

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Maxime Ingremeau. "Local weak limits of Laplace eigenfunctions." Tunisian J. Math. 3 (3) 481 - 515, 2021. https://doi.org/10.2140/tunis.2021.3.481

Information

Received: 2 February 2020; Revised: 9 July 2020; Accepted: 9 August 2020; Published: 2021
First available in Project Euclid: 25 June 2021

Digital Object Identifier: 10.2140/tunis.2021.3.481

Subjects:
Primary: 35P20

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.3 • No. 3 • 2021
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