On the Singular Weyl-Titchmarsh Function of Perturbed Spherical Schroedinger Operators

TL;DR

This paper studies the singular Weyl-Titchmarsh m-function for perturbed spherical Schrödinger operators, establishing existence, properties, and its classification within the generalized Nevanlinna class, linking to super singular perturbations.

Contribution

It provides a detailed analysis of the m-function's properties and its classification, extending the understanding of perturbed Bessel operators with specific perturbations.

Findings

Existence of a fundamental solution system that is entire in the energy parameter.

The singular m-function belongs to the generalized Nevanlinna class.

Connections established between the m-function and super singular perturbation theory.

Abstract

We investigate the singular Weyl-Titchmarsh m-function of perturbed spherical Schroedinger operators (also known as Bessel operators) under the assumption that the perturbation $q(x)$ satisfies $x q(x) \in L^1(0,1)$. We show existence plus detailed properties of a fundamental system of solutions which are entire with respect to the energy parameter. Based on this we show that the singular m-function belongs to the generalized Nevanlinna class and connect our results with the theory of super singular perturbations.