A relative trace formula for obstacle scattering

Abstract

We consider the case of scattering by several obstacles in ${\mathbb{R}}^d$ for $d\ge 2$. In this setting, the absolutely continuous part of the Laplace operator Δ with Dirichlet boundary conditions and the free Laplace operator ${\mathrm{\Delta }}_0$ are unitarily equivalent. For suitable functions that decay sufficiently fast, we have that the difference $g(\mathrm{\Delta })-g({\mathrm{\Delta }}_0)$ is a trace-class operator and its trace is described by the Krein spectral shift function. In this article, we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles, we consider the Laplace operators ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ obtained by imposing Dirichlet boundary conditions only on one of the objects. Our main result in this case states that then $g(\mathrm{\Delta })-g({\mathrm{\Delta }}_1)-g({\mathrm{\Delta }}_2)+g({\mathrm{\Delta }}_0)$ is a trace-class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman–Krein formula. In case $g(x)=x^{\frac{1}{2}}$, the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators. Such integrals have been derived in the physics literature using nonrigorous path integral derivations and our formula provides both a rigorous justification as well as a generalization.