  We generalize earlier studies on the Laplacian for a bounded open domain
$\Omega\in \real^2$ with connected complement and piecewise smooth boundary. We
compare it with the quantum mechanical scattering operator for the exterior of
this same domain. Using single layer and double layer potentials we can prove a
number of new relations which hold when one chooses {\em independently}
Dirichlet or Neumann boundary conditions for the interior and exterior problem.
This relation is provided by a very simple set of $\zeta$-functions, which
involve the single and double layer potentials. We also provide Krein spectral
formulas for all the cases considered and give a numerical algorithm to compute
the $\zeta$-function.
