Schroedinger 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 Schroedinger 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 finite- and infinite-series eigenfunctions. Finite series occur only for special sets of parameters and characterize the quasi-exact solvability. Infinite series occur for all admissible values of the parameters (even values involving finite series), 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 finite series there is a convergent infinite series.