Localization of low-energy eigenfunctions in Šeba billiards

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.