On Realization of the Original Weyl–Titchmarsh Functions by Shrödinger L-systems

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 Shrodinger 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 Shrodinger operator is non-negative. In addition to that we state and prove the uniqueness realization criteria for Shrodinger L-systems with equal boundary parameters. A condition for two Shrodinger 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.