  We consider the case of scattering of several obstacles in $\mathbb{R}^d$ for
$d \geq 2$. In this setting the absolutely continuous part of the Laplace
operator $\Delta$ with Dirichlet boundary conditions and the free Laplace
operator $\Delta_0$ are unitarily equivalent. For suitable functions that decay
sufficiently fast we have that the difference $g(\Delta)-g(\Delta_0)$ is a
trace-class operator and its trace is described by the Krein spectral shift
function. In this paper 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 $\Delta_1$ and $\Delta_2$
obtained by imposing Dirichlet boundary conditions only on one of the objects.
Our main result in this case states that then $g(\Delta) - g(\Delta_1) -
g(\Delta_2) + g(\Delta_0)$ 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 non-rigorous path integral derivations and our formula
provides both a rigorous justification as well as a generalisation.
