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 = -d(2)/dx(2) + q that acts in the space L-2(0, infinity) 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, infinity) 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.