Relative Traces in Obstacle Scattering and the Casimir Effect

25.06.2021, 11:00  –  Online Seminar
Arbeitsgruppenseminar Analysis

Florian Hanisch (Uni Potsdam)

In obstacle scattering, one is interested in properties of the Laplacian ∆ on the complement of a compact set O ("the obstacles") in Euclidean space, subject to certain boundary conditions. ∆ may be compared with the free Laplacian ∆₀ and it is known that differences f(∆) -  f(∆₀) are trace class operators if f satisfies appropriate assumptions. The corresponding traces are given by integrals of the Krein spectral shift function associated with O.

We will discuss a relative version of this result. Assuming that O has several connected components, we look at the setting where all obstacles are present relative to the situation, where all except one have been removed. In this situation, a similar expression for traces can be established which is valid for a much larger class of functions f. We will also explain how these expressions arise in physics as relative (Casimir) energy densities, which are obtained by choosing f to be the square root.

This is joint work with Alden Waters and Alexander Strohmaier.

Forthcoming speakers are Bernadette Lessel on July 2nd, Ihsane Malass on July  9th, Diego López on July 16th, Larisa Jonke on July 23rd.
You are welcome to suggest speakers or topics for the forthcoming sessions.

You are welcome to invite your friends and colleagues to join us! If you wish to attend the talks,  please contact Sylvie Paycha paycha@math.uni-potsdam.de for the login details.