Perturbation determinant and Levinson's formula for Schr\"odinger operators with generalized point interaction

TL;DR

This paper derives a trace formula and explicit perturbation determinant for 1D Schrödinger operators with generalized point interactions, establishing a Levinson's formula analog and analyzing spectral properties.

Contribution

It introduces a new explicit expression for the perturbation determinant and extends Levinson's formula to operators with generalized point interactions.

Findings

Derived a trace formula using the Wronskian.

Expressed the spectral shift function via the perturbation determinant.

Proved an analog of Levinson's formula under certain conditions.

Abstract

We consider the one dimensional Schr\"odinger operator with properly connecting generalized point interaction at the origin. We derive a trace formula for trace of difference of resolvents of perturbed and unperturbed Schr\"odinger operators in terms of a Wronskian which results into an explicit expression for perturbation determinant. Using the estimate for large time real argument on the trace norm of the resolvent difference of the perturbed and unperturbed Schr\"odinger operators we express the spectral shift function in terms of perturbation determinant. Under certain integrability condition on the potential function, we calculate low energy asymptotics for the perturbation determinant and prove an analog of Levinson's formula.