Trace singularities in obstacle scattering and the Poisson relation for the relative trace

TL;DR

This paper investigates the spectral and wave-trace properties of obstacle scattering in Euclidean space, relating singularities in the spectral shift function to geometric features of obstacle configurations and their physical implications.

Contribution

It establishes the analyticity of the Fourier transform of the relative spectral shift near zero and links the decay of a complex function to the first wave-trace invariant for obstacle pairs.

Findings

The Fourier transform of the relative spectral shift function is real-analytic near zero.

Decay properties of the spectral function relate to the shortest obstacle orbit.

Results have implications for Casimir force calculations.

Abstract

We consider the case of scattering of several obstacles in $\mathbb{R}^d$ for $d \geq 2$ for the Laplace operator $\Delta$ with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators $\Delta_1$ and $\Delta_2$ obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative trace operator $g(\Delta) - g(\Delta_1) - g(\Delta_2) + g(\Delta_0)$ was introduced in [18] and shown to be trace-class for a large class of functions $g$, including certrain functions of polynomial growth. When $g$ is sufficiently regular at zero and fast decaying at infinity then, by the Birman-Krein formula, this trace can be computed from the relative spectral shift function $\xi_{rel}(\lambda) = -\frac{1}{\pi} \Im(\Xi(\lambda))$, where $\Xi(\lambda)$ is holomorphic in the upper half-plane and fast decaying. In this paper we study the wave-trace contributions to the singularities of the Fourier transform of $\xi_{rel}$. In particular we prove that $\hat\xi_{rel}$ is real-analytic near zero and we relate the decay of $\Xi(\lambda)$ along the imaginary axis to the first wave-trace invariant of the shortest bounding ball orbit between the obstacles. The function $\Xi(\lambda)$ is important in physics as it determines the Casimir interactions between the objects.