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Geometrical structure of Laplacian eigenfunctions. (English) Zbl 1290.35157

This review article focuses on the classical eigenvalue problem \(-\triangle u_m(x) = \lambda_m u_m(x)\) for \(x \in \Omega\) with Dirichlet, Neumann, or Robin boundary condition, where \(\Omega \subset \mathbb{R}^d\) with piecewise boundary, \(d \geq 2\). As the authors stated in the introduction, one of the goals of this review is to bring together various facts about Laplacian eigenvalues and eigenfunctions and to present them at “a level accessible to scientists from various disciplines.” The main emphasis is put on the geometric structure of Laplacian eigenfunctions and on their relation to the shape of a domain. The contents are: section 1- Introduction. Section 2- Basic properties. Section 3- Eigenbasis for simple Domains. Section 4- Eigenvalues. Section 5- Nodal Lines. Section 6- Estimates for Laplacian Eigenfunctions. Section 7- Localization of Eigenfunctions. Section 8- Further Points and Concluding Remarks. Among them, section 7 is the main section, it was devoted to the spatial structure of eigenfunctions, with special emphasis on their localization in small subsets of a domain. In section 7, the topics being reviewed are: bound quantum states in a potential; Anderson localization; trapping in infinite waveguides; exponential estimate for eigenfunctions; dumbbell domains; localization in irregularly shaped domains; high frequency localization (including whispering gallery modes and focusing modes; bouncing ball modes; domains without localization; quantum billards)

MSC:

35P05 General topics in linear spectral theory for PDEs
35P99 Spectral theory and eigenvalue problems for partial differential equations
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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