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 $hatxi_{rel}$ 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.
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