Schrödinger equation as a confluent Heun equation

Abstract

This article deals with two classes of quasi-exactly solvable (QES) trigonometric potentials for which the one-dimensional Schrödinger equation reduces to a confluent Heun equation (CHE) where the independent variable takes only finite values. Power series for the CHE are used to get polynomial and nonpolynomial eigenfunctions. Polynomials occur only for special sets of parameters and characterize the quasi-exact solvability. Nonpolynomial solutions occur for all admissible values of the parameters (even for values which give polynomials), and are bounded and convergent in the entire range of the independent variable. Moreover, throughout the article we examine other QES trigonometric and hyperbolic potentials. In all cases, for a polynomial solution there is a convergent nonpolynomial solution.