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

Abstract:

The notion of the wave spectrum of a semibounded 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 involves a dynamical system associated with the operator: the wave spectrum is constructed from its reachable sets. In the paper, a description is given for the wave spectrum of the operator $L_0=-\frac {d^2}{dx^2}+q$ that acts in the space $L_2(0,\infty )$ and has defect indices $(1,1)$. A functional (wave) model is constructed for the operator $L_0^*$ in which the elements of the original $L_2(0,\infty )$ are realized as functions on the wave spectrum. This model turns out to be identical to the original $L_0^*$. The latter is fundamental in solving inverse problems: the wave model is determined by their data, which allows reconstruction of the original.