Exact solutions of a quartic potential

Abstract

We find that the analytical solutions to quantum system with a quartic potential [Formula: see text] (arbitrary [Formula: see text] and [Formula: see text] are real numbers) are given by the triconfluent Heun functions [Formula: see text]. The properties of the wave functions, which are strongly relevant for the potential parameters [Formula: see text] and [Formula: see text], are illustrated. It is shown that the wave functions are shrunk to the origin for a given [Formula: see text] when the potential parameter [Formula: see text] increases, while the wave peak of wave functions is concaved to the origin when the negative potential parameter [Formula: see text] increases or parameter [Formula: see text] decreases for a given negative potential parameter [Formula: see text]. The minimum value of the double well case ([Formula: see text]) is given by [Formula: see text] at [Formula: see text].