One-dimensional mappings of the eigenelements of the Schrodinger problem

An approach is proposed for analyzing the inverse spectral problems for the Schrodinger equation based on writing the equation for the analog of the number-of-quanta operator for a harmonic oscillator. This equation makes it possible to determine not only the one-dimensional mapping of the energy eigenvalues but also the linear equation for the point spectrum shift operator of the Schrodinger problem. The solvability conditions of the latter lead to a nonlinear equation that determines the class of allowable potentials. Two classes of potentials regular in R(1) and symmetrical are isolated on the basis of the proposed approach. The first of these leads to equidistant spectra with a gap of arbitrary size and location. The spectrum of the second potential class is a combination of three rigorously equidistant spectra with ground states that are shifted by an arbitrary amount. Generalizations to the case of essentially nonequidistant spectra are shown to be possible.