On localisation of eigenfunctions of the Laplace operator

TL;DR

This paper establishes geometric conditions under which eigenfunctions of the Laplace operator localize in certain domains, providing new insights into eigenfunction behavior in complex geometries.

Contribution

It introduces a simple geometric criterion for eigenfunction localization and demonstrates localization in elongating horn-shaped and polygonal domains.

Findings

Eigenfunctions localize under specific geometric conditions

Localization occurs in horn-shaped domains with elongation

Examples include polygonal domains with Dirichlet or Neumann conditions

Abstract

We prove (i) a simple sufficient geometric condition for localisation of a sequence of first Dirichlet eigenfunctions provided the corresponding Dirichlet Laplacians satisfy a uniform Hardy inequality, and (ii) localisation of a sequence of first Dirichlet eigenfunctions for a wide class of elongating horn-shaped domains. We give examples of sequences of simply connected, planar, polygonal domains for which the corresponding sequence of first eigenfunctions with either Dirichlet, or Neumann, boundary conditions $\kappa$-localise in $L^2$.