Schr\"odinger operators with singular rank-two perturbations and point interactions

TL;DR

This paper constructs norm resolvent approximations for a broad class of point interactions in one-dimensional Schr"odinger operators with singular rank-two perturbations, revealing a rich set of limit operators with various boundary conditions.

Contribution

It introduces new approximation methods for point interactions, including a four-parametric family of connected interactions and various separated boundary conditions, expanding understanding of singular perturbations.

Findings

Limit operators include both connected and separated boundary conditions.

Constructed approximation for a four-parametric subfamily of connected point interactions.

Examples of operators converging to interactions with non-trivial phase parameters.

Abstract

Norm resolvent approximation for a wide class of point interactions in one dimension is constructed. To analyse the limit behaviour of Schr\"odinger operators with localized singular rank-two perturbations coupled with {\delta}-like potentials as the support of perturbation shrinks to a point, we show that the set of limit operators is quite rich. Depending on parameters of the perturbation, the limit operators are described by both the connected and separated boundary conditions. In particular an approximation for a four-parametric subfamily of all the connected point interactions is built. We give examples of the singular perturbed Schr\"odinger operators without localized gauge fields, which converge to point interactions with the non-trivial phase parameter. We also construct an approximation for the point interactions that are described by different types of the separated boundary conditions such as the Robin-Dirichlet, the Neumann-Neumann or the Robin-Robin types.