Traces formulae for relatively Hilbert–Schmidt perturbations

The main goal of this paper is to introduce tools able to replace Krein's spectral shift function when it cannot be defined. We introduce a function η, for a class of perturbations of pseudodifferential operators on $\mathbb{R}$ ^d, when these perturbation decay as 〈x〉^−ρ, ρ>d/2, at infinity. In Euclidean scattering, we establish a complete asymptotic expansion for η, under the usual nontrapping assumption and we deduce a Levinson formula for the Schrödinger operator on $\mathbb{R}$ ³. Some results on the general case ρ>0 are also given.