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 $\mathcal{H}$. 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},\ & Γ_1 u(t) = f(t), &&0\leqslant t \leqslant T, \end{align} where $\{\mathcal{H};{\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2\}$ (${\mathrm{\Gamma }}_{1,2}:\mathcal{H}\to \mathrm{K}\mathrm{e}\mathrm{r}{L}_0^{\ast }$) is the canonical (Vishik) boundary triple for $L_0$, $f$ is a boundary control ($\mathrm{K}\mathrm{e}\mathrm{r}{L}_0^{\ast }$-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 $e\in \mathrm{K}\mathrm{e}\mathrm{r}{L}_0^{\ast }$. Let the map $W^T:\phi \mapsto u^f(T)$ be such that $C^T=(W^T{)}^{\ast }{W}^T=\mathbb{I}+K^T$ 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 Schrödinger operator $S_0=-D^2+q$ in $L_2(0,\mathrm{\infty })$ 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 $L_0$.