Traces formulae for relatively Hilbert–Schmidt perturbations

Article type: Research Article

Authors: Bouclet, Jean‐Marc

Affiliations: Centre for Mathematical Analysis and its Applications, University of Sussex, Falmer, Brighton BN1 9QH, UK E‐mail: j.m.bouclet@sussex.ac.uk

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.

Journal: Asymptotic Analysis, vol. 32, no. 3-4, pp. 257-291, 2002