The Schrodinger particle on the half-line with an attractive δ-interaction: bound states and resonances

In this paper, we provide a detailed description of the eigenvalue ED(x0)<= 0 (respectively, EN(x0)<= 0) of the self-adjoint Hamiltonian operator representing the negative Laplacian on the positive half-line with a Dirichlet (resp. Neuman) boundary condition at the origin perturbed by an attractive Dirac distribution -lambda delta (x-x0) for any fixed value of the magnitude of the coupling constant. We also investigate the lambda -dependence of both eigenvalues for any fixed value of x0. Furthermore, we show that both systems exhibit resonances as poles of the analytic continuation of the resolvent. These results will be connected with the study of the ground state energy of two remarkable three-dimensional self-adjoint operators, studied in depth in Albeverio's et al. monograph, perturbed by an attractive delta -distribution supported on the spherical shell of radius r0.

Automated summary

This paper provides a detailed description of the eigenvalues of the self-adjoint Hamiltonian operator representing the negative Laplacian on the positive half-line with Dirichlet or Neuman boundary conditions perturbed by an attractive Dirac distribution, and investigates their dependence on coupling constant lambda and position parameter x0. The paper also shows resonances as poles of the analytic continuation of the resolvent in both systems, connecting these results to the study of the ground state energy of three-dimensional self-adjoint operators perturbed by attractive delta-distributions on a spherical shell.