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.
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Dedicated with great pleasure to Willi Plessas on the occasion of his 60th birthday.
Based upon work partially supported by the US National Science Foundation under Grant Nos. DMS-0400639 andFRG-0456306.
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Gesztesy, F., Mitrea, M. & Zinchenko, M. On Dirichlet-to-Neumann Maps, Nonlocal Interactions, and Some Applications to Fredholm Determinants. Few-Body Syst 47, 49–64 (2010). https://doi.org/10.1007/s00601-009-0065-0
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DOI: https://doi.org/10.1007/s00601-009-0065-0