The third exactly solvable hypergeometric quantum-mechanical potential

Abstract

We introduce the third independent exactly solvable hypergeometric potential, after the Eckart and the Pöschl-Teller potentials, which is proportional to an energy-independent parameter and has a shape that is independent of this parameter. The general solution of the Schrödinger equation for this potential is written through fundamental solutions each of which presents an irreducible combination of two Gauss hypergeometric functions. The potential is an asymmetric step-barrier with variable height and steepness. Discussing the transmission above such a barrier, we derive a compact formula for the reflection coefficient.

References in corpus (2)

• Thirty five classes of solutions of the quantum time-dependent two-state problem in terms of the general Heun functions
• Expansions of the solutions of the confluent Heun equation in terms of the incomplete Beta and the Appell generalized hypergeometric functions

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• Exact solutions of the sextic oscillator from the bi-confluent Heun equation
• A singular Lambert-W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions
• Generalized hypergeometric solutions of the Heun equation
• Generalized-hypergeometric solutions of the general Fuchsian linear ODE having five regular singularities
• Hermite function solutions of the Schrödinger equation for the sextic oscillator
• A note on the generalized-hypergeometric solutions of general and single-confluent Heun equations
• A conditionally integrable bi-confluent Heun potential involving inverse square root and centrifugal barrier terms
• Darboux partners of Heun-class potentials for the two-dimensional massless Dirac equation
• Two-dimensional Dirac fermion in presence of an asymmetric vector potential
• A new exactly integrable hypergeometric potential for the Schrödinger equation