On realization of the original Weyl-Titchmarsh functions by Shr\"odinger L-systems

TL;DR

This paper explores how Weyl-Titchmarsh functions associated with Schrödinger operators can be realized as impedance functions of specific L-systems, providing explicit constructions, properties, and criteria for uniqueness and shared operators.

Contribution

It provides explicit realizations of Weyl-Titchmarsh functions as impedance functions of Schrödinger L-systems, including criteria for uniqueness and shared main operators.

Findings

Realizations of $-m_ ext{infty}(z)$ and $1/m_ ext{infty}(z)$ as impedance functions.

Explicit L-system constructions related to boundary problems.

Criteria for when two L-systems share the same main operator.

Abstract

We study realizations generated by the original Weyl-Titchmarsh functions $m_\infty(z)$ and $m_\alpha(z)$. It is shown that the Herglotz-Nevanlinna functions $(-m_\infty(z))$ and $(1/m_\infty(z))$ can be realized as the impedance functions of the corresponding Shr\"odinger L-systems sharing the same main dissipative operator. These L-systems are presented explicitly and related to Dirichlet and Neumann boundary problems. Similar results but related to the mixed boundary problems are derived for the Herglotz-Nevanlinna functions $(-m_\alpha(z))$ and $(1/m_\alpha(z))$. We also obtain some additional properties of these realizations in the case when the minimal symmetric Shr\"odinger operator is non-negative. In addition to that we state and prove the uniqueness realization criteria for Shr\"odinger L-systems with equal boun\-dary parameters. A condition for two Shr\"odinger L-systems to share the same main operator is established as well. Examples that illustrate the obtained results are presented in the end of the paper.