Fritz Gesztesy, Marius Mitrea, and Maxim Zinchenko
Variations on a Theme of Jost and Pais
(164K, LaTeX)
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 $\Omega\subset\bbR^n$, $n\in\bbN$,
$n\geq 2$, where $\Omega$ has a compact, nonempty boundary
$\partial\Omega$ satisfying certain regularity conditions. Our
variant involves ratios of perturbation determinants corresponding to
Dirichlet and Neumann boundary conditions on $\partial\Omega$ 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
$L^2(\Omega; d^n x)$, $n\in\bbN$, to modified Fredholm determinants
associated with operators in $L^2(\partial\Omega; d^{n-1}\sigma)$,
$n\geq 2$.
Applications involving the Birman--Schwinger principle and eigenvalue
counting functions are discussed.