The Dirichlet Laplacian with a point interaction on unbounded Lipschitz domains

Abstract

We study one-centre point interactions for the Dirichlet Laplacian on unbounded domains in dimensions two and three, with emphasis on exterior domains and special Lipschitz domains. These operators are singular perturbations constructed as self-adjoint extensions of the Dirichlet Laplacian restricted to functions vanishing at the interaction centre, and their resolvents are given by an explicit Kre\uın formula with a single extension parameter $α$. The negative spectrum is completely characterized by a scalar equation and the critical coupling $α$ separating binding from non-binding is the threshold limit of the Weyl function appearing in the Kre\uın formula. We establish domain monotonicity of the Weyl function, of the critical coupling, and of the unique negative eigenvalue when existing, and we derive sharp near-boundary asymptotics of the critical coupling in uniformly $C^{1,1}$ geometries. These estimates imply that, for every fixed coupling, nonpositive spectrum disappears when the interaction centre approaches the Dirichlet boundary. We also prove limiting absorption principles and purely absolutely continuous positive spectrum for a point-interaction in exterior domains case and in classes of special Lipschitz domains. We also analyze in depth several threshold phenomena. We show that the critical coupling is governed by the far-field behavior of the zero-energy Green function: exterior domains give threshold resonances, domains contained in a three-dimensional half-space give threshold eigenvalues, the half-plane gives a $p$-wave resonance, and planar wedges exhibit types of threshold states that are aperture-dependent. Finally, low-energy resolvent expansions are computed in the model cases and persistence or disappearance of eigenvalues at threshold are studied. The present paper seems to be the first systematic work on the subject.