### 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.

Original language | English |
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Article number | 1650177 |

Journal | Modern Physics Letters A |

Volume | 31 |

Issue number | 33 |

DOIs | |

Publication status | Published - 30 Oct 2016 |

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### Keywords

- confluent hypergeometric function
- double-confluent Heun equation
- integrable potentials
- Stationary Schrödinger equation

### ASJC Scopus subject areas

- Nuclear and High Energy Physics
- Astronomy and Astrophysics

### Cite this

*Modern Physics Letters A*,

*31*(33), [1650177]. https://doi.org/10.1142/S0217732316501777

**A singular Lambert-W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions.** / Ishkhanyan, A. M.

Research output: Contribution to journal › Article

*Modern Physics Letters A*, vol. 31, no. 33, 1650177. https://doi.org/10.1142/S0217732316501777

}

TY - JOUR

T1 - A singular Lambert-W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions

AU - Ishkhanyan, A. M.

PY - 2016/10/30

Y1 - 2016/10/30

N2 - 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.

AB - 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.

KW - confluent hypergeometric function

KW - double-confluent Heun equation

KW - integrable potentials

KW - Stationary Schrödinger equation

UR - http://www.scopus.com/inward/record.url?scp=84991219687&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84991219687&partnerID=8YFLogxK

U2 - 10.1142/S0217732316501777

DO - 10.1142/S0217732316501777

M3 - Article

AN - SCOPUS:84991219687

VL - 31

JO - Modern Physics Letters A

JF - Modern Physics Letters A

SN - 0217-7323

IS - 33

M1 - 1650177

ER -