Hyperbolic-type potentials from the confluent Heun equation: symmetric double-well potentials

Abstract Based on the confluence properties of a second-order differential equation applied to the general Heun’s equation, an appropriate form of the confluent Heun differential equation is obtained. Then, under a general transformation derived from an auxiliary function, the confluent Heun equation is transformed into a Schrödinger-like equation. Such auxiliary function allows finding the appropriate change of variable to arrive at a Schrödinger equation with a multiparameter hyperbolic-type potential. Under an appropriate selection of the involved parameters, families of symmetric double-well potential are obtained. It is shown that a particular form of the parameters gives place to the Downing potential, which is a double-well potential given in terms of hyperbolic functions. Likewise, the proposal can be extended to the case of asymmetric double-well potentials and triple-well ones that could be useful in physics and quantum chemistry.