The Schr\"odinger particle on the half-line with an attractive $\delta$-interaction: bound states and resonances

TL;DR

This paper analyzes the spectral properties of a Schrödinger particle on the half-line with an attractive delta interaction, detailing bound states, eigenvalues, and resonances, and connecting these findings to three-dimensional models with spherical delta potentials.

Contribution

It provides a comprehensive description of eigenvalues and resonances for the half-line Schrödinger operator with delta interaction, including their dependence on parameters and connections to 3D models.

Findings

Eigenvalues are explicitly characterized for Dirichlet and Neumann boundary conditions.

Resonances are identified as poles of the analytically continued resolvent.

Results connect one-dimensional delta interactions with three-dimensional spherical shell models.

Abstract

In this paper we provide a detailed description of the eigenvalue $ E_{D}(x_0)\leq 0$ (respectively $ E_{N}(x_0)\leq 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 $-\lambda \delta(x-x_0)$ 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 $x_0$. 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 monograph, perturbed by an attractive $\delta$-distribution supported on the spherical shell of radius $r_0$.