Spectral Theory and Resonances for Point Interactions in Unbounded Domains

In this work we aim to give a first attempt at describing spectral properties and resonances of the Laplacian perturbed by point interaction on unbounded domains. In the first chapter, which is preparatory but still contains not previously published material, we define the N-point interactions in an unbounded domain, with Dirichlet, Neumann or Robin boundary conditions using the theory of self-adjoint extensions. We also give a different treatment through a different road and, following a work Teta, where he defines quadratic forms corresponding to the point interaction Laplacians in R^n, we construct quadratic forms corresponding to delta-interactions in unbounded domains. The second chapter is devoted to the study of the spectrum of 1-point interaction in unbounded domains. The point spectrum in these domains is constituted at most of an eigenvalue, due to Krein formula. The existence and behavior of the eigenvalue is determined by two elements: the value of the parameter alpha and the position of the point interaction with respect to the boundary. General conditions are found for exterior domains in R^3 and R^2 with Dirichlet boundary conditions and for domains \Omega, star-shaped with respect to the point in which the "delta intersction" is located, for Neumann boundary conditions, that include the half-space and other domains with unbounded boundary. Then the critical alpha for which an eigenvalue exists is determined, in some particular cases in which the Green's function is known explicitly such as the half-space, the half-plane, the exterior of the disk and the exterior of a sphere. The behavior of alpha_ textup c changes in a noticeable way when compared to the one in R^ n for a single interaction. In the third chapter we define resonances for point interactions in domains (according to a definition agreeing with the Sjostrand-Zworski theory for regular perturbation of the laplacian and of their more general "black-box" definition) and study their distribution on the half-space and half-plane for 1-point interaction. In the former case, for Dirichlet and Neumann boundary conditions, we determine that there are infinite resonances (while for N=1 in R^ 3 there is at most one resonance). Moreover each of this resonances is localized in a precise vertical strip of the complex plane. Also a logarithmic relation between real and imaginary part of the resonances is established. As a byproduct of this, an asymptotic on the number of resonances in a ball of radius R centered at the origin for R to+ \infty is performed. For Robin boundary condition in the half-space and Dirichlet and Neumann ones in the half-plane, we give an estimate on the real part of the resonances contained in a horizontal strip of width \beta. As a non-trivial application we give the resonance expansion for the wave evolution. Finally, we studied the semiclassical distribution of resonances for the one-point interaction on the half-space. A previous result in the one dimensional case of a delta interaction placed on the half-line with Dirichlet boundary condition at the free end (also called the Winter model in physical literature) on the half-line. The present results constitute a first generalization of this analysis to a three dimensional case.