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Local weak limits of
Laplace eigenfunctions
Maxime Ingremeau |
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Vol. 3 (2021), No. 3, 481–515
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Abstract |
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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 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
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Mathematical Subject Classification 2010
Primary: 35P20
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Milestones
Received: 2 February 2020
Revised: 9 July 2020
Accepted: 9 August 2020
Published: 13 May 2021
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Authors |
| Maxime Ingremeau | |
| Laboratoire J.A. Dieudonné UMR
CNRS Université Côte d’Azur Nice France |
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