A characterization of singular Schrödinger operators on the half-line

Abstract

We study a class of delta-like perturbations of the Laplacian on the half-line, characterized by Robin boundary conditions at the origin. Using the formalism of nonstandard analysis, we derive a simple connection with a suitable family of Schrödinger operators with potentials of very large (infinite) magnitude and very short (infinitesimal) range. As a consequence, we also derive a similar result for point interactions in the Euclidean space ${\mathbb{R}}^3$, in the case of radial potentials. Moreover, we discuss explicitly our results in the case of potentials that are linear in a neighbourhood of the origin.

16 pages

References in corpus (4)

• On point interactions realised as Ter-Martirosyan-Skornyakov Hamiltonians
• The $L^p$ boundedness of wave operators for the three-dimensional multi-centre point interaction
• Fractional powers and singular perturbations of quantum differential Hamiltonians
• Infinitesimal and Infinite Numbers as an Approach to Quantum Mechanics