TTMS15. On Weyl Titchmarsh functions and Schrodinger L-systems
Speaker: Dr. Sergey Belyi (Troy University)
Abstract. The focus of this talk is set on classical objects in Schr\"odinger operators theory, the original Weyl-Titchmarsh functions and their connections with conservative L-systems with Schr\"odinger operators. 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 Schr\"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 Schr\"odinger operator is non-negative. Examples that illustrate the obtained results are presented in the end of the presentation.