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

Abstract

We study L-system realizations of the original Weyl-Titchmarsh functions $(-m_{\alpha }(z))$. In the case when the minimal symmetric Schrödinger operator is non-negative, we describe the Schrödinger 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.

17 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:2006.13045, arXiv:1709.07370, arXiv:2108.06567, arXiv:2012.08069