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Multi-dimensional versions of a determinant formula due to Jost and Pais

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Title: Multi-dimensional versions of a determinant formula due to Jost and Pais
Authors: Gesztesy, F.1 http://www.math.missouri.edu/personnel/faculty/gesztesyf.html, Mitrea, M.1 http://www.math.missouri.edu/personnel/faculty/mitream.html, Zinchenko, M.2 maxim@caltech.edu
Source: Reports on Mathematical Physics. Jun2007, Vol. 59 Issue 3, p365-377. 13p.
Subjects: Determinants (Mathematics), Boundary value problems, Astronomical perturbation, Algebra
Abstract: We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schrödinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is replaced by an open set Ω ⊂ ℝ n , n = 2, 3, where Ω has a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on ∂Ω and invokes the corresponding Dirichlet-to-Neumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants perturbation associated with operators in L 2 (Ω d n x) to modified Fredholm determinants associated with operators in L 2(∂Ω; d n−lσ), n = 2, 3. [Copyright &y& Elsevier]
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Database: Engineering Source
Description
Abstract:We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schrödinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is replaced by an open set Ω ⊂ ℝ n , n = 2, 3, where Ω has a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on ∂Ω and invokes the corresponding Dirichlet-to-Neumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants perturbation associated with operators in L 2 (Ω d n x) to modified Fredholm determinants associated with operators in L 2(∂Ω; d n−lσ), n = 2, 3. [Copyright &y& Elsevier]
ISSN:00344877
DOI:10.1016/S0034-4877(07)80072-3