1D Schr\"{o}dinger operators with short range interactions: two-scale regularization of distributional potentials

TL;DR

This paper studies the limiting behavior of 1D Schrödinger operators with short-range, distributional potentials, showing how different scaling limits lead to various point interactions or decoupled operators.

Contribution

It introduces a two-scale regularization approach for distributional potentials in 1D Schrödinger operators and characterizes the limit operators based on potential shapes and scaling ratios.

Findings

Norm resolvent convergence of operators as psilon, nu  0

Limit operators depend on potential shape and scaling ratio

Zero-energy resonance leads to non-trivial point interactions

Abstract

For real bounded functions \Phi and \Psi of compact support, we prove the norm resolvent convergence, as \epsilon and \nu tend to 0, of a family of one-dimensional Schroedinger operators on the line of the form S_{\epsilon, \nu}= -D^2+\alpha\epsilon^{-2}\Phi(\epsilon^{-1}x)+\beta\nu^{-1}\Psi(\nu^{-1}x), provided the ratio \nu/\epsilon has a finite or infinity limit. The limit operator S_0 depends on the shape of \Phi and \Psi as well as on the limit of ratio \nu/\epsilon. If the potential \alpha\Phi possesses a zero-energy resonance, then S_0 describes a non trivial point interaction at the origin. Otherwise S_0 is the direct sum of the Dirichlet half-line Schroedinger operators.