Abstract:

Spectral theory of the Laplace operator on unbounded domains in R^n is conveniently described by stationary scattering theory. One of the major results in this theory is that traces of differences of functions of perturbed and unperturbed operator are related by the spectral shift function. The Birman-Krein formula then allows to express this spectral shift function by the scattering matrix. We show that in the context of many obstacles a relative spectral shift function can be defined that applies to a significantly larger class of functions including unbounded functions of polynomial growth. This generalises formulae that have been used in QFT to compute Casimir forces and that were derived using path integrals. I will give the notations and the setup and describe the main theorem. If there is time I will explain the relation to QFT and the applications of the theorem. (joint work with F. Hanisch and A. Waters)