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 $\mathrm{\Omega }\subset {\text{bbR}}^n$, $n\in \text{bbN}$, $n\geq 2$,where$\Omega$hasacompact,nonemptyboundary$\partial\Omega$ satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on $\mathrm{\partial }\mathrm{\Omega }$ 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 $L^2(\mathrm{\Omega };d^{n}x)$, $n\in \text{bbN}$, to modified Fredholm determinants associated with operators in $L^2(\mathrm{\partial }\mathrm{\Omega };d^{n-1}\sigma )$, $n\ge 2$. Applications involving the Birman-Schwinger principle and eigenvalue counting functions are discussed.