Local Weak Limits of Laplace Eigenfunctions

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.

• Publication year

  2017

• Venue

  arXiv: Analysis of PDEs

• Publication date

  2017-12-09

• Fields of study

  Mathematics, Physics

• Identifiers
  DOI 10.2140/tunis.2021.3.481arXiv 1712.03431
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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