Meromorphic continuation of the spectral shift function

Abstract

We obtain a representation of the derivative of the spectral shift function $\xi(\lambda,h)$ in the framework of semiclassical "black box" perturbations. Our representation implies a meromorphic continuation of $\xi(\lambda,h)$ involving the semiclassical resonances. Moreover, we obtain a Weyl-type asymptotics of the spectral shift function, as well as a Breit-Wigner approximation in an interval $(\lambda -\delta,\lambda+\delta), 0<\delta<\epsilon h$.