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- Title
On Dirichlet-to-Neumann Maps, Nonlocal Interactions, and Some Applications to Fredholm Determinants.
- Authors
Gesztesy, Fritz; Mitrea, Marius; Zinchenko, Maxim
- Abstract
We consider Dirichlet-to-Neumann maps associated with (not necessarily self-adjoint) Schrödinger operators describing nonlocal interactions in $${L^2(\Omega; d^n x)}$$ , where $${\Omega \subset \mathbb{R}^n}$$ , $${n\in\mathbb{N}}$$ , $${n\geq 2}$$ , are open sets with a compact, nonempty boundary $${\partial\Omega}$$ 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 $${L^2(\partial\Omega; d^{n-1} \sigma)}$$ , $${n\in\mathbb{N}}$$ , $${n\geq 2}$$ . 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 Schrödinger operator on the half-line $${(0,\infty)}$$ , in the case of local interactions, to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation.
- Subjects
FREDHOLM equations; SCHRODINGER equation; WRONSKIAN determinant; DIFFERENTIAL equations; DIRICHLET problem
- Publication
Few-Body Systems, 2010, Vol 47, Issue 1-2, p49
- ISSN
0177-7963
- Publication type
Article
- DOI
10.1007/s00601-009-0065-0