Weyl-Titchmarsh Theory for Schroedinger Operators with Strongly Singular Potentials

TL;DR

This paper extends Weyl-Titchmarsh theory to Schrödinger operators with strongly singular potentials, providing criteria for spectral analysis, integral representations, and uniqueness results, especially for perturbed spherical Schrödinger operators.

Contribution

It introduces new criteria for the analytic extension of the singular Weyl function and derives integral representations and spectral support criteria for strongly singular potentials.

Findings

Criteria for analytic extension of the singular Weyl function.

Integral representation of the singular Weyl function.

Spectral support determination from boundary behavior.

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.