Variations on a Theme of Jost and Pais

Abstract

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 $\Omega \subset {\mathbb{R}}^n$, $n\in \mathbb{N}$, $n\ge 2$, 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 $L^2(\Omega ;d^{n}x)$, $n\in \mathbb{N}$, to modified Fredholm determinants associated with operators in $L^2(\partial \Omega ;d^{n-1}\sigma )$, $n\ge 2$. Applications involving the Birman-Schwinger principle and eigenvalue counting functions are discussed.

40 pages. To appear in J. Funct. Anal