Schödinger operators on half-line with shrinking potentials at the origin
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Date
2015
Journal Title
Journal ISSN
Volume Title
Publisher
SISSA
Abstract
We discuss the general model of a Schrödinger quantum particle
constrained on a straight half-line with given self-adjoint boundary condition at the origin and an interaction potential supported around the origin. We study the limit when the range of the potential scales to zero and its magnitude blows up. We show that in the limit the dynamics is generated by a self-adjoint negative Laplacian on the half-line, with a possible preservation or modification of the boundary condition at the origin, depending on the magnitude of the scaling and of the strength of the potential.