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 Schrodinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrodinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is replaced by an open set Omega subset of R-n, n is an element of N, n >= 2, where Omega has a compact, nonempty boundary partial derivative Omega satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on a partial derivative 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 is an element of N, to modified Fredholm determinants associated with operators in L-2(partial derivative Omega; d(n-l) sigma), n >= 2. Applications involving the Birman-Schwinger principle and eigenvalue counting functions are discussed. (c) 2007 Elsevier Inc. All rights reserved.