Wave model of the Sturm-Liouville operator on the half-line

The notion of the wave spectrum of a semi-bounded symmetric operator was introduced by one of the authors in 2013. The wave spectrum is a topological space determined by the operator in a canonical way. The definition uses a dynamical system associated with the operator: the wave spectrum is constructed from its reachable sets. In the paper we give a description of the wave spectrum of the operator $L_0=-\frac{d^2}{d{x}^2}+q$ which acts in the space $L_2(0,\mathrm{\infty })$ and has defect indices $(1,1)$. We construct a functional (wave) model of the operator $L_0^{\ast }$ in which the elements of the original $L_2(0,\mathrm{\infty })$ are realized as functions on the wave spectrum. It turns out to be identical to the original $L_0^{\ast }$. The latter is fundamental in solving inverse problems: the wave model is determined by their data, which allows for reconstruction of the original.