Eigenvalues of \v{S}eba billiards with localization of low-energy eigenfunctions

TL;DR

This paper investigates how a point scatterer on a thin rectangle causes low-energy eigenfunctions to localize, using explicit spectral analysis and numerical simulations to understand the influence of the rectangle's aspect ratio.

Contribution

It provides an explicit characterization of localized eigenfunctions in Seba billiards and links their behavior to a one-dimensional Schrödinger operator with a delta potential.

Findings

Eigenfunctions localize to specific parts of the rectangle.

Localization rate depends on the rectangle's aspect ratio.

Numerical results support the asymptotic analysis.

Abstract

We study the localization of eigenfunctions produced by a point scatterer on a thin rectangle. We find an explicit set of eigenfunctions localized to part of the rectangle by showing that the one-dimensional Schr\"odinger operator with a Dirac delta potential asymptotically governs the spectral properties of the two-dimensional point scatterer. We also find the rate of localization in terms of the aspect ratio of the rectangle. In addition, we present numerical results regarding the asymptotic behavior of the localization.