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arXiv · 2311.01612

A functional model of a class of symmetric semi-bounded operators

M. I. Belishev·S. A. Simonov
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Abstract

Let $L_0$ be a closed symmetric positive definite operator with nonzero defect indices $n_\pm(L_0)$ in a separable Hilbert space ${\mathscr H}$. It determines a family of dynamical systems $\alpha^T$, $T>0$, of the form \begin{align*} & u"(t)+L_0^*u(t) = 0 && {\rm in}\,\,\,{{\mathscr H}}, \,\,\,0 0$. We show that under these assumptions the operator $L_0$ is unitarily equivalent to the minimal Schr\"{o}dinger operator $S_0=-D^2+q$ in ${L_2(0,\infty)}$ with a smooth real-valued potential $q$, which is in the limit point case at infinity. It is also proved that $S_0$ provides a canonical wave model of $L_0$.

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M. I. Belishev, S. A. Simonov. 2023-11-02. A functional model of a class of symmetric semi-bounded operators. https://arxiv.org/abs/2311.01612

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