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

29 pages, 1 figure. arXiv admin note: text overlap with arXiv:1709.07370

Cited by in corpus (1)

• The original Weyl-Titchmarsh functions and sectorial Shrödinger L-systems