The original Weyl-Titchmarsh functions and sectorial Shr\"odinger L-systems

TL;DR

This paper explores the realization of Weyl-Titchmarsh functions through L-systems associated with Schrödinger operators, establishing conditions for sectorial classes and geometric structures of these systems.

Contribution

It introduces new conditions for representing Weyl-Titchmarsh functions as impedance functions of Schrödinger L-systems and describes their geometric and sectorial properties.

Findings

Characterization of $(-m_eta(z))$ as impedance functions

Conditions for $(-m_eta(z))$ to belong to sectorial classes

Description of the geometric structure of L-systems based on parameters

Abstract

In this paper we study the L-system realizations generated by the original Weyl-Titchmarsh functions $m_\alpha(z)$ in the case when the minimal symmetric Shr\"o\-dinger operator in $L_2[\ell,+\infty)$ is non-negative. We realize functions $(-m_\alpha(z))$ as impe\-dance functions of Shr\"odinger L-systems and derive necessary and sufficient conditions for $(-m_\alpha(z))$ to fall into sectorial classes $S^{\beta_1,\beta_2}$ of Stieltjes functions. Moreover, it is shown that the knowledge of the value $m_\infty(-0)$ and parameter $\alpha$ allows us to describe the geometric structure of the L-system that realizes $(-m_\alpha(z))$. Conditions when the main and state space operators of the L-system realizing $(-m_\alpha(z))$ have the same or not angle of sectoriality are presented in terms of the parameter $\alpha$. Example that illustrates the obtained results is presented in the end of the paper.