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.


Dr. Anna Skripka

Research Area

Pure Maths Seminar


Texas A&M University


Mon, 13/07/2009 - 12:00pm