The third five-parametric hypergeometric quantum-mechanical potential

Abstract

We introduce the third five-parametric ordinary hypergeometric energy-independent quantum-mechanical potential, after the Eckart and Pöschl-Teller potentials, which is proportional to an arbitrary variable parameter and has a shape that is independent of that parameter. Depending on an involved parameter, the potential presents either a short-range singular well (which behaves as inverse square root at the origin and vanishes exponentially at infinity) or a smooth asymmetric step-barrier (with variable height and steepness). The general solution of the Schrödinger equation for this potential, which is a member of a general Heun family of potentials, is written through fundamental solutions each of which presents an irreducible linear combination of two Gauss ordinary hypergeometric functions.

References in corpus (5)

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• One-dimensional Schrödinger equation with non-analytic potential $V(x)=-g^2\exp (-|x|)$ and its exact Bessel-function solvability
• Morse potential, symmetric Morse potential and bracketed bound-state energies
• A singular Lambert-W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions
• Symmetrized exponential oscillator