### Abstract

We show that there exist 35 choices for the coordinate transformation each leading to a potential for which the stationary Schrödinger equation is exactly solvable in terms of the general Heun functions. Because of the symmetry of the Heun equation with respect to the transposition of its singularities only eleven of these potentials are independent. Four of these independent potentials are always explicitly written in terms of elementary functions, one potential is given through the Jacobi elliptic sn-function, and the others are in general defined parametrically. Nine of the independent potentials possess exactly or conditionally integrable hypergeometric sub-potentials for which each of the fundamental solutions of the Schrödinger equation is written through a single hypergeometric function. Many of the potentials possess sub-potentials for which the general solution is written through fundamental solutions each of which is a linear combination of two or more Gauss hypergeometric functions. We present an example of such a potential which is a conditionally integrable generalization of the third exactly solvable Gauss hypergeometric potential.

Original language | English |
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Pages (from-to) | 456-471 |

Number of pages | 16 |

Journal | Annals of Physics |

Volume | 388 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

### Fingerprint

### Keywords

- General Heun equation
- Integrable potential
- Stationary Schrödinger equation

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Annals of Physics*,

*388*, 456-471. https://doi.org/10.1016/j.aop.2017.11.033

**Schrödinger potentials solvable in terms of the general Heun functions.** / Ishkhanyan, A. M.

Research output: Contribution to journal › Article

*Annals of Physics*, vol. 388, pp. 456-471. https://doi.org/10.1016/j.aop.2017.11.033

}

TY - JOUR

T1 - Schrödinger potentials solvable in terms of the general Heun functions

AU - Ishkhanyan, A. M.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - We show that there exist 35 choices for the coordinate transformation each leading to a potential for which the stationary Schrödinger equation is exactly solvable in terms of the general Heun functions. Because of the symmetry of the Heun equation with respect to the transposition of its singularities only eleven of these potentials are independent. Four of these independent potentials are always explicitly written in terms of elementary functions, one potential is given through the Jacobi elliptic sn-function, and the others are in general defined parametrically. Nine of the independent potentials possess exactly or conditionally integrable hypergeometric sub-potentials for which each of the fundamental solutions of the Schrödinger equation is written through a single hypergeometric function. Many of the potentials possess sub-potentials for which the general solution is written through fundamental solutions each of which is a linear combination of two or more Gauss hypergeometric functions. We present an example of such a potential which is a conditionally integrable generalization of the third exactly solvable Gauss hypergeometric potential.

AB - We show that there exist 35 choices for the coordinate transformation each leading to a potential for which the stationary Schrödinger equation is exactly solvable in terms of the general Heun functions. Because of the symmetry of the Heun equation with respect to the transposition of its singularities only eleven of these potentials are independent. Four of these independent potentials are always explicitly written in terms of elementary functions, one potential is given through the Jacobi elliptic sn-function, and the others are in general defined parametrically. Nine of the independent potentials possess exactly or conditionally integrable hypergeometric sub-potentials for which each of the fundamental solutions of the Schrödinger equation is written through a single hypergeometric function. Many of the potentials possess sub-potentials for which the general solution is written through fundamental solutions each of which is a linear combination of two or more Gauss hypergeometric functions. We present an example of such a potential which is a conditionally integrable generalization of the third exactly solvable Gauss hypergeometric potential.

KW - General Heun equation

KW - Integrable potential

KW - Stationary Schrödinger equation

UR - http://www.scopus.com/inward/record.url?scp=85040792837&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85040792837&partnerID=8YFLogxK

U2 - 10.1016/j.aop.2017.11.033

DO - 10.1016/j.aop.2017.11.033

M3 - Article

VL - 388

SP - 456

EP - 471

JO - Annals of Physics

JF - Annals of Physics

SN - 0003-4916

ER -