Instructor: Konstantin Makarov

Description: The concept of the spectral shift function in perturbation theory of self-adjoint operators appeared at the beginning of fifties in the physics literature in the papers of Ilya Lifshitz. The spectral shift function arises in the theory of trace class perturbations in connection with the trace formula (an integral representation for the trace of the difference of functions of two self-adjoint operators) which is a far going generalization of the classical Fundamental Theorem of Calculus.

We will describe and correlate several representations of the spectral shift function and plan to discuss a variety of applications of the concept in mathematical physics including scattering theory, spectral theory of random Schröinger operators, and dynamical systems.

The course can be considered an introduction to the modern spectral theory of linear operators and it is intended for students whose interests lie in analysis, mathematical physics, spectral theory, probability theory, and foundations of Quantum Theory.