Resonances and resonance expansions for point interactions on the half-space

TL;DR

This paper characterizes the resonances of a point interaction perturbation of the Laplacian on the half-space, revealing an infinite set with a specific asymptotic distribution and applications to wave and Schrödinger dynamics.

Contribution

It provides a complete characterization of resonances for point interactions on the half-space, including their distribution and semiclassical asymptotics, which was not previously known.

Findings

Resonances form an infinite set with a specific asymptotic distribution.

Resonances satisfy a modified Weyl law.

Applications to wave and Schrödinger dynamics on the half-space.

Abstract

In this paper we describe the resonances of the singular perturbation of the Laplacian on the half space $\Omega =\mathbb R^3_+$ given by the self-adjoint operator named $\delta$-interaction. We will assume Dirichlet or Neumann boundary conditions on $\partial \Omega$. At variance with the well known case of $\mathbb R^3$, the resonances constitute an infinite set, here completely characterized. Moreover, we prove that resonances have an asymptotic distribution satisfying a modified Weyl law and we give the semiclassical asymptotics. Finally we give applications of the results to the asymptotic behavior of the abstract wave and Schr\"odinger dynamics generated by the Laplacian with a point interaction on the half-space