Spectral flow and Levinson's theorem for Schr\"o dinger operators

TL;DR

This paper introduces a novel proof of Levinson's theorem for Schrödinger operators using spectral flow, applicable across all dimensions and accommodating resonances, with results expressed via spectral shift functions.

Contribution

It provides a new, unified proof of Levinson's theorem leveraging spectral flow, valid in all dimensions and with resonances, expanding previous methods.

Findings

Proof valid in all dimensions

Applicable in presence of resonances

Expressed via spectral shift function

Abstract

We use spectral flow to present a new proof of Levinson's theorem for Schr\"{o}dinger operators on $\mathbb{R}^n$ with smooth compactly supported potential. Our proof is valid in all dimensions and in the presence of resonances. The statement is expressed in terms of the spectral shift function and the ``high energy corrected time delay'' following Guillop\'{e} and others.