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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]
Copyright of Reports on Mathematical Physics is the property of Pergamon Press - An Imprint of Elsevier Science and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: Multi-dimensional versions of a determinant formula due to Jost and Pais
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  Data: <searchLink fieldCode="AR" term="%22Gesztesy%2C+F%2E%22">Gesztesy, F.</searchLink><relatesTo>1</relatesTo><i> http://www.math.missouri.edu/personnel/faculty/gesztesyf.html</i><br /><searchLink fieldCode="AR" term="%22Mitrea%2C+M%2E%22">Mitrea, M.</searchLink><relatesTo>1</relatesTo><i> http://www.math.missouri.edu/personnel/faculty/mitream.html</i><br /><searchLink fieldCode="AR" term="%22Zinchenko%2C+M%2E%22">Zinchenko, M.</searchLink><relatesTo>2</relatesTo><i> maxim@caltech.edu</i>
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  Data: <searchLink fieldCode="JN" term="%22Reports+on+Mathematical+Physics%22">Reports on Mathematical Physics</searchLink>. Jun2007, Vol. 59 Issue 3, p365-377. 13p.
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  Data: <searchLink fieldCode="DE" term="%22Determinants+%28Mathematics%29%22">Determinants (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Boundary+value+problems%22">Boundary value problems</searchLink><br /><searchLink fieldCode="DE" term="%22Astronomical+perturbation%22">Astronomical perturbation</searchLink><br /><searchLink fieldCode="DE" term="%22Algebra%22">Algebra</searchLink>
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  Data: 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]
– Name: AbstractSuppliedCopyright
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  Group: Ab
  Data: <i>Copyright of Reports on Mathematical Physics is the property of Pergamon Press - An Imprint of Elsevier Science and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.1016/S0034-4877(07)80072-3
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        Text: English
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      – SubjectFull: Boundary value problems
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      – SubjectFull: Astronomical perturbation
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      – SubjectFull: Algebra
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              Text: Jun2007
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