Spectral distributions for long range perturbations

We study distributions which generalize the concept of spectral shift function, for pseudo-differential operators on Rd. We call such distributions spectral distributions. Relations between relative scattering determinants and spectral distributions are established; they lead to the definition of regularized scattering phase. These relations are analogous to the usual one for the standard spectral shift function. We give several asymptotic properties in the high energy and semiclassical limits where both nontrapping and trapping cases are considered. In particular, we prove Breit–Wigner formulae for the regularized scattering phases, for semiclassical Schrödinger operators with long-range potentials.