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

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 $L_2[\mathrm{\ell },+\mathrm{\infty })$ 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 $m_{\mathrm{\infty }}(-0)$ 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.