Unification and extension of the factorization method for constructing exactly and conditionally-exactly solvable potentials. The case of a single potential generating function

Abstract

The article proposes a new algorithm for applying the factorization method to the problem of calculating the spectrum of exactly and conditionally exactly solvable potentials. The proposed algorithm allows us to unify and extend the capabilities of the factorization method to construct exactly solvable potentials. The new approach is demonstrated by calculating the eigenvalues ​​of exactly solvable potentials constructed using a single function in the form of the Laurent-type polynomial. The algorithm makes it possible to significantly simplify the scheme for calculating the spectrum, parameters of the superpotential, as well as the constrain conditions for the parameters of the potential, in the case of conditionally exactly solvable potentials. It is shown that the shape of the spectrum is determined only by the differential equation, which is satisfied by the potential generating function.