The spectral shift function and Levinson's theorem for quantum star graphs

TL;DR

This paper derives formulas for the spectral shift function and Levinson's theorem for Schrödinger operators on star-shaped quantum graphs, connecting spectral properties with graph structure.

Contribution

It introduces new representations for the spectral shift function and an analog of Levinson's formula specific to quantum star graphs.

Findings

Derived an expression for the trace difference of resolvents using Wronskian

Established formulas for the perturbation determinant and spectral shift function

Presented an analog of Levinson's theorem for star-shaped quantum graphs

Abstract

We consider the Schr\"odinger operator on a star shaped graph with $n$ edges joined at a single vertex. We derive an expression for the trace of the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads to representations for the perturbation determinant and the spectral shift function, and to an analog of Levinson's formula.