Potentials of the Heun class

Abstract

We review different methods of generating potentials such that the one-dimensional Schrödinger equation (ODSE) can be transformed into the hypergeometric equation. We compare our results with previous studies, and complement the subject with new findings. Our main result is to derive new classes of potentials such that the ODSE can be transformed into the Heun equation and its confluent cases. The generalized Heun equation is also considered.

References in corpus (6)

• Heun equation, Teukolsky equation, and type-D metrics
• Exactly solvable Schrödinger operators
• Examples of Heun and Mathieu functions as solutions of wave equations in curved spaces
• Heun's equation, generalized hypergeometric function and exceptional Jacobi polynomial
• The generalized Heun equation in QFT in curved space-times
• Incomplete beta-function expansions of the solutions to the confluent Heun equation

Cited by in corpus (12)

• Solutions of the bi-confluent Heun equation in terms of the Hermite functions
• Schrödinger potentials solvable in terms of the general Heun functions
• Discretization of Natanzon potentials
• The Lambert-W step-potential - an exactly solvable confluent hypergeometric potential
• Expansions of the solutions to the confluent Heun equation in terms of the Kummer confluent hypergeometric functions
• Exact solutions of the sextic oscillator from the bi-confluent Heun equation
• Potentials of the Heun class: the triconfluent case
• A singular Lambert-W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions
• Scalar field in Reissner-Nordström spacetime: Bound state and scattering state
• Expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta and Gamma functions
• A conditionally integrable bi-confluent Heun potential involving inverse square root and centrifugal barrier terms
• Semi-commuting and commuting operators for the Heun family