- 10-101 Aleksey Kostenko, Alexander Sakhnovich, and Gerald Teschl
- Weyl-Titchmarsh Theory for Schroedinger Operators with Strongly Singular Potentials
Jul 1, 10
(auto. generated ps),
of related papers
Abstract. We 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. We will derive an integral
representation for this singular Weyl function and give a criterion when it is a generalized Nevanlinna
function. Moreover, we will show how essential supports for the Lebesgue decomposition of the
spectral measure can be obtained from the boundary behavior of the singular Weyl function.
Finally, we will prove a local Borg-Marchenko type uniqueness result.
Our criteria will in particular cover the aforementioned case
of perturbed spherical Schroedinger operators.