Resonances and resonance expansions for $$\delta $$-interactions on the half-space

Abstract In this paper we describe the resonances of the singular perturbation of the Laplacian on the half-space $$\Omega =\mathbb R^3_+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mi>R</mml:mi> <mml:mo>+</mml:mo> <mml:mn>3</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> given by the self-adjoint operator named $$\delta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>δ</mml:mi> </mml:math> -interaction. We will assume Dirichlet or Neumann boundary conditions on $$\partial \Omega $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> . At variance with the well-known case of $$\mathbb {R}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> , 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 Schrödinger dynamics generated by the Laplacian with a point interaction on the half-space.