Localization of low-energy eigenfunctions in Seba billiards

TL;DR

This paper studies how a point scatterer on a rectangular plate causes low-energy eigenfunctions of the Laplacian to localize on one side, revealing a new confinement phenomenon not seen with Dirichlet conditions.

Contribution

It demonstrates that a point scatterer induces low-energy mode localization on a rectangular plate, a behavior not caused by boundary conditions alone, and extends this understanding to higher modes with increased eccentricity.

Findings

Point scatterer acts as a barrier confining low-energy modes.

Dirichlet boundary condition at a point does not cause localization.

Localization persists in higher modes as the plate's eccentricity increases.

Abstract

We investigate localization of low-energy modes of the Laplacian with a point scatterer on a rectangular plate. We observe that the point scatterer acts as a barrier confining the low-level modes to one side of the plate while assuming the Dirichlet boundary condition at a point does not induce this type of localization. This low-energy phenomenon extends to higher modes as we increase the eccentricity of the plate.