On Dirichlet-to-Neumann Maps, Nonlocal Interactions, and Some Applications to Fredholm Determinants

TL;DR

This paper explores Dirichlet-to-Neumann maps for non-self-adjoint Schrödinger operators with nonlocal interactions, linking Fredholm determinants in bulk and boundary spaces, and extending classical formulas to broader contexts.

Contribution

It introduces a reduction technique connecting Fredholm determinants of operators in domain and boundary spaces, extending the Jost-Pais formula to nonlocal interactions.

Findings

Reduction of Fredholm determinants from domain to boundary operators

Extension of Jost-Pais formula to nonlocal Schrödinger interactions

New analytical tools for non-self-adjoint Schrödinger operators

Abstract

We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrodinger operators describing nonlocal interactions in $L^2(\Omega; d^n x)$, $n\geq 2$, 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 Fredholm perturbation determinants associated with operators in $L^2(\Omega; d^n x)$ to Fredholm perturbation determinants associated with operators in $L^2(\partial\Omega; d^{n-1}\sigma)$. This leads to an extension of a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with a Schr\"odinger operator on the half-line $(0,\infty)$, in the case of local interactions, to a Wronski determinant of appropriate distributional solutions of the underlying Schrodinger equation.