Asymptotics Near ±m of the Spectral Shift Function for Dirac Operators with Non-Constant Magnetic Fields

We consider a 3-dimensional Dirac operator H0 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 (H0+V, H0). We obtain, as a by-product, a generalized version of Levinson's Theorem relating the eigenvalues asymptotics of H0+V near +m and -m to the scattering phase shift for the pair (H0+V, H0).