On Local Perturbations of SCHR\"Odinger Operator on Plane

TL;DR

This paper establishes precise conditions under which small eigenvalues appear for Schrödinger operators on the plane when subjected to local perturbations, and provides asymptotic descriptions of these eigenvalues.

Contribution

It offers necessary and sufficient criteria for eigenvalue emergence and constructs their asymptotics, advancing understanding of spectral changes under local perturbations.

Findings

Derived conditions for eigenvalue emergence.

Constructed asymptotic formulas for small eigenvalues.

Provided illustrative examples.

Abstract

We obtain necessary and sufficient conditions for emerging of small eigenvalue for Schr\"odinger operator on plane under local operator perturbations. In the case the eigenvalue emerges we construct its asymptotics. The examples are given.