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
,
,
n2, 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 associated with operators in
L2(Ω;dnx),
, to modified Fredholm determinants associated with operators in
L2(∂Ω;dn−1σ),
n2. Applications involving the Birman–Schwinger principle and eigenvalue counting functions are discussed.