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

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.