  We investigate the scattering of a point particle from n non-overlapping,
disconnected hard disks which are fixed in the two-dimensional plane and study
the connection between the spectral properties of the quantum-mechanical
scattering matrix and its semiclassical equivalent based on the semiclassical
zeta function of Gutzwiller and Voros. We rewrite the determinant of the
scattering matrix in such a way that it separates into the product of n
determinants of 1-disk scattering matrices - representing the incoherent part
of the scattering from the n disk system - and the ratio of two mutually
complex conjugate determinants of the genuine multi-scattering kernel, M, which
is of Korringa-Kohn-Rostoker-type and represents the coherent multi-disk aspect
of the n-disk scattering. Our result is well-defined at every step of the
calculation, as the on-shell T-matrix and the kernel M-1 are shown to be
trace-class. We stress that the cumulant expansion (which defines the
determinant over an infinite, but trace class matrix) imposes the curvature
regularization scheme to the Gutzwiller-Voros zeta function and thus leads to a
new, well-defined and direct derivation of the semiclassical spectral function.
We show that unitarity is preserved even at the semiclassical level.
