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

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