On the eigenvalue spacing distribution for a point scatterer on the flat torus

TL;DR

This paper investigates how the eigenvalue spacing distribution of a point scatterer on a flat torus compares to the Laplacian spectrum, revealing similar behavior in 2D but different phenomena in 3D.

Contribution

It demonstrates that in 2D, the eigenvalue spacings of a point scatterer match the Laplacian's, while in 3D, the distribution exhibits distinct characteristics.

Findings

Eigenvalue spacings in 2D match Laplacian spectrum

Perturbed eigenvalues cluster with unperturbed ones in 2D

Different eigenvalue behavior observed in 3D

Abstract

We study the level spacing distribution for the spectrum of a point scatterer on a flat torus. In the 2-dimensional case, we show that in the weak coupling regime the eigenvalue spacing distribution coincides with that of the spectrum of the Laplacian (ignoring multiplicties), by showing that the perturbed eigenvalues generically clump with the unperturbed ones on the scale of the mean level spacing. We also study the three dimensional case, where the situation is very different.