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

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

2010-02-02

arxiv.org/abs/1002.0390v2

Mathematics

Spectral Theory

18 pages

Scientific paper

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.

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