Relative Traces in Obstacle Scattering

04.02.2021, 16:15  –  Online-Seminar
Forschungsseminar Differentialgeometrie

Florian Hanisch (UP)

In obstacle scattering, one is interested in properties of the Laplacian
∆ on the complement of a compact set O ("the obstacles") in Euclidean
space. It may be compared with the free Laplacian ∆₀  and it is known
that differences f(∆) -  f(∆₀) are trace class operators if  f satisfies
certain restrictive assumptions. Traces are then 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
two connected components, we look at the setting where both obstacles
are present relative to the situation, where one of them has been
removed. We discuss that in in this situation, a similar expression for
traces of differences can now be established for a much larger class of
functions f. This is important for physical applications where relative
(Casimir) energy densities are obtained by choosing f to be the square
root.  This is joint work with Alden Waters and Alexander Strohmaier.

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