A model and characterization of a class of symmetric semibounded operators

Abstract

Let $\mathcal{G}$ be a Hilbert space and $\mathfrak{B}(\mathcal{G})$ the algebra of bounded operators, $\mathcal{H}=L_2([0,\infty );\mathcal{G})$. An operator-valued function $Q\in L_{\infty ,\mathrm{loc}}\left([0,\infty );\mathfrak{B}(\mathcal{G})\right)$ determines a multiplication operator in $\mathcal{H}$ by $(Qy)(x)=Q(x)y(x)$, $x⩾0$. We say that an operator $L_0$ in a Hilbert space is a Schrödinger type operator, if it is unitarily equivalent to $-d^2/d{x}^2+Q(x)$ on a relevant domain. The paper provides a characterization of a class of such operators. The characterization is given in terms of properties of an evolutionary dynamical system associated with $L_0$. It provides a way to construct a functional Schrödinger model of $L_0$.