A singular Lambert-W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions

Abstract

We introduce two potentials explicitly given by the Lambert-W function for which the exact solution of the one-dimensional stationary Schrödinger equation is written through the first derivative of a double-confluent Heun function. One of these potentials is a singular potential that behaves as the inverse square root in the vicinity of the origin and vanishes exponentially at the infinity. The exact solution of the Schrödinger equation for this potential is given through fundamental solutions each of which presents an irreducible linear combination of two confluent hypergeometric functions. Since the potential is effectively a short-range one it supports only a finite number of bound states.

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Cited by in corpus (11)

• Schrödinger potentials solvable in terms of the general Heun functions
• Discretization of Natanzon potentials
• Exact solution of the Schrödinger equation for a short-range exponential potential with inverse square root singularity
• Exact solutions of the sextic oscillator from the bi-confluent Heun equation
• Hermite function solutions of the Schrödinger equation for the sextic oscillator
• Darboux partners of Heun-class potentials for the two-dimensional massless Dirac equation
• Two-dimensional Dirac fermion in presence of an asymmetric vector potential
• Study of superradiance in the Lambert-W potential barrier
• Semiclassical rates for tunnel ionization from power-law potentials induced by a constant or low-frequency electric field
• A new exactly integrable hypergeometric potential for the Schrödinger equation
• The third five-parametric hypergeometric quantum-mechanical potential