Traces formulae for relatively Hilbert–Schmidt perturbations

Abstract

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.