On Dirichlet-to-Neumann Maps and Some Applications to Modified Fredholm Determinants

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrodinger operators in $L^2(\Omega; d^n x)$, $n=2,3$, 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 modified Fredholm perturbation determinants associated with operators in $L^2(\Omega; d^n x)$ to modified Fredholm perturbation determinants associated with operators in $L^2(\partial\Omega; d^{n-1}\sigma)$, $n=2,3$. This leads to a two- and three-dimensional extension of a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with a Schrodinger operator on the half-line $(0,\infty)$ to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrodinger equation.