A new Levinson's theorem for potentials with critical decay

TL;DR

This paper extends Levinson's theorem to Schrödinger operators with potentials decaying as 1/|x|^2, analyzing low-energy spectral behavior in cases with zero eigenvalues and resonances.

Contribution

It introduces a generalized Levinson's theorem applicable to potentials with critical decay, accounting for zero eigenvalues and resonances.

Findings

Derived a low-energy asymptotic formula for the spectral shift function

Established conditions for zero eigenvalues and resonances in critical decay potentials

Extended Levinson's theorem to include potentials with decay rate O(1/|x|^2)

Abstract

We study the low-energy asymptotics of the spectral shift function for Schr\"odinger operators with potentials decaying like $O(\frac{1}{|x|^2})$. We prove a generalized Levinson's for this class of potentials in presence of zero eigenvalue and zero resonance.