Variations on a Theme of Jost and Pais

We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schr\"odinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schr\"odinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is replaced by an open set $\Omega\subset\bbR^n$, $n\in\bbN$, $n\geq 2$, where $\Omega$ has a compact, nonempty boundary $\partial\Omega$ satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on $\partial\Omega$ and invokes the corresponding Dirichlet-to-Neumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants associated with operators in $L^2(\Omega; d^n x)$, $n\in\bbN$, to modified Fredholm determinants associated with operators in $L^2(\partial\Omega; d^{n-1}\sigma)$, $n\geq 2$. Applications involving the Birman-Schwinger principle and eigenvalue counting functions are discussed.