Realization of Inverse Stieltjes Functions \((-m_\alpha (z))\) by Schrödinger L-Systems

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