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 $Q\in L_{\infty,\rm loc}\left([0,\infty);\mathfrak B(\mathcal G)\right)$ determines a multiplication operator in $\mathcal H$ by $(Qy)(x)=Q(x)y(x)$, $x\geqslant0$. We say that an operator $L_0$ in a Hilbert space is a Schrödinger type operator, if it is unitarily equivalent to $-d^2/dx^2+Q(x)$ 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 $L_0$.

Theorems & Definitions (12)

• Proposition 1
• Definition
• Proposition 2
• Lemma 1
• proof
• Lemma 2
• Theorem 1
• Theorem 2
• proof
• Theorem 3