### Abstract

We construct an expansion of the solutions of the bi-confluent Heun equation in terms of the Hermite functions. The series is governed by a three-term recurrence relation between successive coefficients of the expansion. We examine the restrictions that are imposed on the involved parameters in order that the series terminates thus resulting in closed-form finite-sum solutions of the bi-confluent Heun equation. A physical application of the closed-form solutions is discussed. We present the five six-parametric potentials for which the general solution of the one-dimensional Schrödinger equation is written in terms of the bi-confluent Heun functions and further identify a particular conditionally integrable potential for which the involved bi-confluent Heun function admits a four-term finite-sum expansion in terms of the Hermite functions. This is an infinite well defined on a half-axis. We present the explicit solution of the one-dimensional Schrödinger equation for this potential and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and construct an accurate approximation for the bound-state energy levels.

Original language | English |
---|---|

Pages (from-to) | 79-91 |

Number of pages | 13 |

Journal | Annals of Physics |

Volume | 383 |

DOIs | |

Publication status | Published - 1 Aug 2017 |

### Fingerprint

### Keywords

- Bi-confluent Heun equation
- Hermite function
- Series expansion

### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Annals of Physics*,

*383*, 79-91. https://doi.org/10.1016/j.aop.2017.04.015

**Solutions of the bi-confluent Heun equation in terms of the Hermite functions.** / Ishkhanyan, T. A.; Ishkhanyan, A. M.

Research output: Contribution to journal › Article

*Annals of Physics*, vol. 383, pp. 79-91. https://doi.org/10.1016/j.aop.2017.04.015

}

TY - JOUR

T1 - Solutions of the bi-confluent Heun equation in terms of the Hermite functions

AU - Ishkhanyan, T. A.

AU - Ishkhanyan, A. M.

PY - 2017/8/1

Y1 - 2017/8/1

N2 - We construct an expansion of the solutions of the bi-confluent Heun equation in terms of the Hermite functions. The series is governed by a three-term recurrence relation between successive coefficients of the expansion. We examine the restrictions that are imposed on the involved parameters in order that the series terminates thus resulting in closed-form finite-sum solutions of the bi-confluent Heun equation. A physical application of the closed-form solutions is discussed. We present the five six-parametric potentials for which the general solution of the one-dimensional Schrödinger equation is written in terms of the bi-confluent Heun functions and further identify a particular conditionally integrable potential for which the involved bi-confluent Heun function admits a four-term finite-sum expansion in terms of the Hermite functions. This is an infinite well defined on a half-axis. We present the explicit solution of the one-dimensional Schrödinger equation for this potential and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and construct an accurate approximation for the bound-state energy levels.

AB - We construct an expansion of the solutions of the bi-confluent Heun equation in terms of the Hermite functions. The series is governed by a three-term recurrence relation between successive coefficients of the expansion. We examine the restrictions that are imposed on the involved parameters in order that the series terminates thus resulting in closed-form finite-sum solutions of the bi-confluent Heun equation. A physical application of the closed-form solutions is discussed. We present the five six-parametric potentials for which the general solution of the one-dimensional Schrödinger equation is written in terms of the bi-confluent Heun functions and further identify a particular conditionally integrable potential for which the involved bi-confluent Heun function admits a four-term finite-sum expansion in terms of the Hermite functions. This is an infinite well defined on a half-axis. We present the explicit solution of the one-dimensional Schrödinger equation for this potential and discuss the bound states supported by the potential. We derive the exact equation for the energy spectrum and construct an accurate approximation for the bound-state energy levels.

KW - Bi-confluent Heun equation

KW - Hermite function

KW - Series expansion

UR - http://www.scopus.com/inward/record.url?scp=85019997001&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85019997001&partnerID=8YFLogxK

U2 - 10.1016/j.aop.2017.04.015

DO - 10.1016/j.aop.2017.04.015

M3 - Article

AN - SCOPUS:85019997001

VL - 383

SP - 79

EP - 91

JO - Annals of Physics

JF - Annals of Physics

SN - 0003-4916

ER -