Asymptotics near +-m of the spectral shift function for Dirac operators with non-constant magnetic fields

TL;DR

This paper analyzes the asymptotic behavior of the spectral shift function for 3D Dirac operators with non-constant magnetic fields near the spectral edges, establishing a generalized Levinson's Theorem relating eigenvalues to scattering phases.

Contribution

It provides new asymptotic formulas for the spectral shift function near spectral edges for Dirac operators with non-constant magnetic fields, extending Levinson's Theorem.

Findings

Asymptotic formulas for spectral shift function near ±m

Generalized Levinson's Theorem relating eigenvalues and scattering phases

Results applicable to Dirac operators with non-constant magnetic fields

Abstract

We consider a 3-dimensional Dirac operator H_0 with non-constant magnetic field of constant direction, perturbed by a sign-definite matrix-valued potential V decaying fast enough at infinity. Then we determine asymptotics, as the energy goes to +m and -m, of the spectral shift function for the pair (H_0,H_0+V). We obtain, as a by-product, a generalised version of Levinson's Theorem relating the eigenvalues asymptotics of H_0+V near +m and -m to the scattering phase shift for the pair (H_0,H_0+V).