Realization of inverse Stieltjes functions $(-m_\alpha(z))$ by Schrodinger L-systems

We study L-system realizations of the original Weyl-Titchmarsh functions $(-m_\alpha(z))$. In the case when the minimal symmetric Schr\"odinger operator is non-negative, we describe the Schr\"odinger L-systems that realize inverse Stieltjes functions $(-m_\alpha(z))$. This approach allows to derive a necessary and sufficient conditions for the functions $(-m_\alpha(z))$ to be inverse Stieltjes. In particular, the criteria when $(-m_\infty(z))$ is an inverse Stieltjes function is provided. Moreover, the value $m_\infty(-0)$ and parameter $\alpha$ allow us to describe the geometric structure of the realizing $(-m_\alpha(z))$ L-system. Additionally, we present the conditions in terms of the parameter $\alpha$ when the main and associated operators of a realizing $(-m_\alpha(z))$ L-system have the same or different angle of sectoriality which sets connections with the Kato problem on sectorial extensions of sectorial forms.