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

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<t<T,\\ & u(0)=u'(0)=0 && {\rm in}\,\,\,{{\mathscr H}},\\ & \Gamma_1 u(t) = f(t), &&0\leqslant t \leqslant T, \end{align*} where $\{{\mathscr H};\Gamma_1,\Gamma_2\}$ ($\Gamma_{1,2}:{\mathscr H}\to{\rm Ker\,} L_0^*$) is the canonical (Vishik) boundary triple for $L_0$, $f$ is a boundary control (${\rm Ker\,} L_0^*$-valued function of $t$) and $u=u^f(t)$ is the solution (trajectory). Let $L_0$ be completely non-self-adjoint and $n_\pm(L_0)=1$, so that $f(t)=\phi(t)e$ with a scalar function $\phi\in {L_2(0,T)}$ and $e\in{\rm Ker\,} L_0^*$. Let the map $W^T: \phi\mapsto u^f(T)$ be such that $C^T=(W^T)^*W^T=\mathbb I+K^T$ with an integral operator $K^T$ in ${L_2(0,T)}$ which has a smooth kernel. Assume that $C^T$ an isomorphism in ${L_2(0,T)}$ for all $T>0$. 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 $L_0$.