On Dirichlet-to-Neumann Maps, Nonlocal Interactions, and Some Applications to Fredholm Determinants

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrodinger operators describing nonlocal interactions in $L^2(\mathrm{\Omega };d^{n}x)$, $n\ge 2$, where $\mathrm{\Omega }$ 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 $L^2(\mathrm{\Omega };d^{n}x)$ 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 $(0,\mathrm{\infty })$, in the case of local interactions, to a Wronski determinant of appropriate distributional solutions of the underlying Schrodinger equation.