Variations on a Theme of Jost and Pais

TL;DR

This paper extends a formula relating Fredholm perturbation determinants for Schrödinger operators from one dimension to higher dimensions, involving boundary conditions and Dirichlet-to-Neumann maps, with applications to eigenvalue analysis.

Contribution

It generalizes a known one-dimensional determinant reduction formula to higher dimensions using boundary operators and Dirichlet-to-Neumann maps.

Findings

Reduced ratios of Fredholm determinants to boundary operators.

Connected perturbation determinants with boundary conditions.

Discussed applications to eigenvalue counting and Birman-Schwinger principle.

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