Spectral shift functions

Abstract: 

Spectral shift functions (SSF) provide information about a quantitative change of the spectrum of a self-adjoint operator under the influence of a self-adjoint perturbation. The first order SSF, called Krein's SSF, was introduced in [Lifshits '52 + Krein '53]. Since then, it has been well explored and found in various problems of mathematical physics; it can also be recognized as the scattering phase [Birman, Krein '62] and the spectral flow in a non-commutative geometry setting [Azamov, Carey, Sukochev '07]. The first order SSF can govern only the case of a trace class perturbation (or a trace class difference of the resolvents). In order to encompass more general perturbations, one needs to consider modified, higher order, spectral shift functions. The second order SSF is due to [Koplienko '84]; the spectral shift functions of order greater than two are under development. The talk will give a brief overview of the spectral shift functions, including recent results on the ones of order greater than two, obtained by the speaker in collaboration with K. Dykema.