  We consider Dirichlet-to-Neumann maps associated with (not necessarily
self-adjoint) Schrodinger operators describing nonlocal interactions in
$L^2(\Omega; d^n x)$, $n\geq 2$, where $\Omega$ is an open set with a compact,
nonempty boundary satisfying certain regularity conditions. As an application
we describe a reduction of a certain ratio of Fredholm perturbation
determinants associated with operators in $L^2(\Omega; d^n x)$ to Fredholm
perturbation determinants associated with operators in $L^2(\partial\Omega;
d^{n-1}\sigma)$. This leads to an extension of a variant of a celebrated
formula due to Jost and Pais, which reduces the Fredholm perturbation
determinant associated with a Schr\"odinger operator on the half-line
$(0,\infty)$, in the case of local interactions, to a Wronski determinant of
appropriate distributional solutions of the underlying Schrodinger equation.
