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Variations on a Theme of Jost and Pais | | Fritz Gesztesy
; Marius Mitrea
; Maxim Zinchenko
; | | Date: |
24 May 2007 | | Subject: | Spectral Theory (math.SP); Mathematical Physics (math-ph) | | 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 Schr"odinger operator on a half-line to a simple Wronski determinant of
appropriate distributional solutions of the underlying Schr"odinger equation,
generalizes to higher dimensions. In this multi-dimensional extension the
half-line is replaced by an open set $OmegasubsetbR^n$, $ninbN$, $ngeq
2$, where $Omega$ has a compact, nonempty boundary $partialOmega$ satisfying
certain regularity conditions. Our variant involves ratios of perturbation
determinants corresponding to Dirichlet and Neumann boundary conditions on
$partialOmega$ and invokes the corresponding Dirichlet-to-Neumann map. As a
result, we succeed in reducing a certain ratio of modified Fredholm
perturbation determinants associated with operators in $L^2(Omega; d^n x)$,
$ninbN$, to modified Fredholm determinants associated with operators in
$L^2(partialOmega; d^{n-1}sigma)$, $ngeq 2$.
Applications involving the Birman-Schwinger principle and eigenvalue counting
functions are discussed. | | Source: | arXiv, arxiv.0705.3510 | | Services: | Forum | Review | PDF | Favorites | |
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