Alexander Sakhnovich
Weyl-Titchmarsh Theory for Schroedinger Operators with Strongly Singular Potentials.
Fri, 07/02/2010 - 15:01 — SomNambulAuthors: Aleksey Kostenko, Alexander Sakhnovich, Gerald Teschl
Subjects: Spectral Theory
AbstractWe develop Weyl-Titchmarsh theory for Schroedinger operators with strongly
singular potentials such as perturbed spherical Schroedinger operators (also
known as Bessel operators). It is known that in such situations one can still
define a corresponding singular Weyl m-function and it was recently shown that
there is also an associated spectral transformation. Here we will give a
general criterion when the singular Weyl function can be analytically extended
to the upper half plane.Construction of the solution of the inverse spectral problem for a system depending rationally on the spectral parameter, Borg-Marchenko-type theorem, and sine-Gordon equation.
Fri, 01/15/2010 - 16:01 — SomNambulAuthors: Alexander Sakhnovich
Subjects: Classical Analysis and ODEs
AbstractWeyl theory for a non-classical system depending rationally on the spectral
parameter is treated. Borg-Marchenko-type uniqueness theorem is proved. The
solution of the inverse problem is constructed. An application to sine-Gordon
equation in laboratory coordinates is given.On the GBDT version of the B\"acklund-Darboux transformation and its applications to the linear and nonlinear equations and Weyl theory.
Wed, 09/09/2009 - 22:08 — SomNambulAuthors: Alexander Sakhnovich
Subjects: Classical Analysis and ODEs
AbstractA general theorem on the GBDT version of the B\"acklund-Darboux
transformation for systems rationally depending on the spectral parameter is
treated and its applications to nonlinear equations are given. Explicit
solutions of direct and inverse problems for Dirac-type systems, including
systems with singularities, and for the system auxiliary to the $N$-wave
equation are reviewed. New results on explicit construction of the wave
functions for radial Dirac equation are obtained.
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