Potentials of the Heun class: the triconfluent case

Abstract

Since the advent of quantum mechanics different approaches to find analytical solutions of the Schrödinger equation have been successfully developed. Here we follow and generalize the approach pioneered by Natanzon and others by which the Schrödinger equations can be transformed into another well-known equation for transcendental function (e.g., the hypergeometric equation). This sets a class of potentials for which this transformation is possible. Our generalization consists in finding potentials allowing the transformation of the Schrödinger equation into a triconfluent Heun equation. We find the energy eigenvalues of this class of potentials, the eigenfunction and the exact superpartners.

arXiv comment18 pages, 5 figures

References in corpus (4)

• Exactly solvable Schrödinger operators
• Potentials of the Heun class
• Anharmonic Oscillator Equations:Treatment Parallel to Mathieu Equation
• On the Liouville transformation and exactly-solvable Schrodinger equations

Cited by in corpus (8)

• Exact solution of the Schrödinger equation for the inverse square root potential $V_0/\sqrt{x}$
• Discretization of Natanzon potentials
• A conditionally exactly solvable generalization of the inverse square root potential
• Expansions of the solutions of the general Heun equation governed by two-term recurrence relations for coefficients
• Hermite function solutions of the Schrödinger equation for the sextic oscillator
• Scalar field in Reissner-Nordström spacetime: Bound state and scattering state
• A conditionally integrable bi-confluent Heun potential involving inverse square root and centrifugal barrier terms
• Semi-commuting and commuting operators for the Heun family