On Dirichlet-to-Neumann Maps and Some Applications to Modified Fredholm Determinants

TL;DR

This paper explores Dirichlet-to-Neumann maps for non-self-adjoint Schrödinger operators and connects Fredholm determinants in the domain to those on the boundary, extending classical formulas to higher dimensions.

Contribution

It introduces a reduction technique linking Fredholm determinants of Schrödinger operators in the domain to boundary operators, extending the Jost-Pais formula to 2D and 3D.

Findings

Reduction of Fredholm determinants from domain to boundary operators

Extension of Jost-Pais formula to higher dimensions

Connection between Dirichlet-to-Neumann maps and Fredholm determinants

Abstract

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrodinger operators in $L^2(\Omega; d^n x)$, $n=2,3$, where $\Omega$ is an open set with a compact, nonempty boundary satisfying certain regularity conditions. As an application we describe a reduction of a certain ratio of modified Fredholm perturbation determinants associated with operators in $L^2(\Omega; d^n x)$ to modified Fredholm perturbation determinants associated with operators in $L^2(\partial\Omega; d^{n-1}\sigma)$, $n=2,3$. This leads to a two- and three-dimensional extension of a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with a Schrodinger operator on the half-line $(0,\infty)$ to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrodinger equation.