Concentration and confinement of eigenfunctions in a bounded open set (version 2)

TL;DR

This paper investigates how eigenfunctions of a layered media operator concentrate or disperse within a bounded domain, providing conditions to distinguish between localized and delocalized eigenfunctions and correcting previous results.

Contribution

It introduces a partition of eigenfunctions based on their concentration properties, offering new criteria and representations for layered media operators, with improved proofs and generalizations.

Findings

Identifies conditions for eigenfunction concentration and dispersion.

Provides a spectral representation related to layered media.

Corrects and simplifies previous proofs and results.

Abstract

Consider the Dirichlet-Laplacian in $\Omega:= (0,L)\times (0,H)$ and choose another open set $\omega\subset \Omega$. The estimate $0<C_{\omega}\leq R_{\omega}(u):=\frac{\Vert u\Vert^{2}_{L^{2}(\omega)}}{\Vert u\Vert^{2}_{L^{2}(\Omega)}}\leq \frac{vol(\omega)}{vol(\omega)}$, for all the eigenfunctions, is well known. This is no longer true for an inhomogeneous elliptic selfadjoint operator $A$. In this work we create a partition among the set of eigenfunctions: $\forall \omega$, the eigenfunctions satisfy $R_{omega}>C_{\omega}>0,\exists \omega, \omega\not=\emptyset$, such that $\inf R_{\omega}(u)=0$,and we wish to characterize these two sets. For two patterns we give a sufficient condition, sometimes necessary. As our operator corresponds to a layered media we can give another representation of its spectrum: i.e. a subset of points of $R\times R$ that leads to the suggested partition and others connected results: micro local interpretation, default measures,... Section 4.1 of the previous version was not correct, now it is corrected, many proofs are simplified and a new general result is added.