Equidistant spectra of anharmonic oscillators

Some representative potentials of the anharmonic‐oscillator type are constructed. Some corresponding spectra‐shift operators are also constructed. These operators are a natural generalization of Fok creation and annihilation operators. The Schrödinger problem for these potentials leads to an equidistant energy spectrum for all excited states, which are separated from the ground state by an energy gap. The general properties of the dynamic system generated by spectral‐shift operators of third degree are analyzed. Several examples of such anharmonic oscillators are discussed. The relationship between the eigenvectors of the Schrödinger problem and a certain type of nonclassical orthogonal polynomials is established.