Multi-dimensional Versions of a Determinant Formula Due to 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 Schrödinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation, generalizes to higher dimensions. In this multi-dimensional extension, the half-line is replaced by an open set Ω ⊂ Rn, n = 2, 3, where Ω has a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on ∂Ω and invokes the corresponding Dirichlet-to-Neumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants perturbation associated with operators in L2(Ω; dnx) to modified Fredholm determinants associated with operators in L2(∂Ω; dn−1σ), n = 2, 3.