Resonances and Spectral Shift Function for the semi-classical Dirac operator

We consider the self-adjoint operator $H=H_0+V$, where $H_0$ is the free semi-classical Dirac operator on $R^3$. We suppose that the smooth matrix-valued potential $V=O(<x>^{-\delta}), \delta>0,$ has an analytic continuation in a complex sector outside a compact. We define the resonances as the eigenvalues of the non-selfadjoint operator obtained from the Dirac operator $H$ by a complex distortions of $R^{3}$.We establish an upper bound $O(h^{-3})$ for the number of resonances in any compact domain. For $\delta>3$, a representation of the derivative of the spectral shift function $\xi(\lambda,h)$ related to the semi-classical resonances of $H$ and a local trace formula are obtained. In particular, if $V$ is an electro-magnetic potential, we deduce a Weyl-type asymptotic of the spectral shift function. As a by-product, we obtain an upper bound $O(h^{-2})$ for the number of resonances close to non-critical energy levels in domains of width $h$ and a Breit-Wigner approximation formula for the derivative of the spectral shift function.