The Lambert-W step-potential - an exactly solvable confluent hypergeometric potential

Abstract

We present an asymmetric step-barrier potential for which the one-dimensional stationary Schrödinger equation is exactly solved in terms of the confluent hypergeometric functions. The potential is given in terms of the Lambert -function, which is an implicitly elementary function also known as the product logarithm. We present the general solution of the problem and consider the quantum reflection at transmission of a particle above this potential barrier. Compared with the abrupt-step and hyperbolic tangent potentials, which are reproduced by the Lambert potential in certain parameter and/or variable variation regions, the reflection coefficient is smaller because of the lesser steepness of the potential on the particle incidence side. Presenting the derivation of the Lambert potential we show that this is a four-parametric sub-potential of a more general five-parametric one also solvable in terms of the confluent hypergeometric functions. The latter potential, however, is a conditionally integrable one. Finally, we show that there exists one more potential the solution for which is written in terms of the derivative of a bi-confluent Heun function.

Phys. Lett. A (2016)

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• Exact solution of the Schrödinger equation for the inverse square root potential $V_0/\sqrt{x}$
• Thirty five classes of solutions of the quantum time-dependent two-state problem in terms of the general Heun functions
• Discretization of Natanzon potentials
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• A singular Lambert-W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions
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• A new exactly integrable hypergeometric potential for the Schrödinger equation
• The third five-parametric hypergeometric quantum-mechanical potential
• Exactly-solvable quantum systems in terms of Lambert-W functions