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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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