Journal of Operator Theory

Volume 96, Issue 1, Summer 2026 pp. 257-275.

A model and characterization of a class of symmetric semibounded operators

Authors: Mikhail I. Belishev (1), Sergey A. Simonov (2)
Author institution: (1) St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka 27, St. Petersburg, Russia, 191023
(2) St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Fontanka 27, St. Petersburg, Russia, 191023 and Alferov Academic University of the Russian Academy of Sciences, Khlopina 8/3, St. Petersburg, Russia

Summary: Let ${\mathscr G}$ be a Hilbert space, let ${\mathfrak B}({\mathscr G})$ be the algebra of bounded operators, and let $\widetilde{\mathscr H}=L_2([0,\infty);{\mathscr G})$. An operator-valued function $Q\in L_{\infty,\rm loc} ([0,\infty);{\mathfrak B}({\mathscr G}) )$ determines a multiplication operator in $\widetilde{\mathscr H}$ by $(Qy)(x)=Q(x)y(x)$, $x\geqslant0$. We say that an operator $L_0$ in a Hilbert space $\mathscr H$ is a Schr\"odinger type operator, if it is unitarily equivalent to $S=-({\mathrm d^2}/{\mathrm dx^2})+Q$, which acts in $\widetilde{\mathscr H}$ on a relevant domain. The paper provides a characterization of a class of such operators $L_0$. The characterization is given in terms of the properties of an evolutionary dynamical system associated with $L_0$. It is constructive: a way to realize $L_0$ in the form of $S$ is proposed.

DOI: http://dx.doi.org/10.7900/jot.2025apr05.2488
Keywords: boundary triple, dynamical system, functional model, triangular factorization, inverse problem, characterization, matrix Schr\"odinger operator

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