### Abstract

In this paper, the sextic oscillator is discussed as a potential obtained from the bi-confluent Heun equation after a suitable variable transformation. Following earlier results, the solutions of this differential equation are expressed as a series expansion of Hermite functions with shifted and scaled arguments. The expansion coefficients are obtained from a three-term recurrence relation. It is shown that this construction leads to the known quasi-exactly solvable (QES) form of the sextic oscillator when some parameters are chosen in a specific way. By forcing the termination of the recurrence relation, the Hermite functions turn into Hermite polynomials with shifted arguments, and, at the same time, a polynomial expression is obtained for one of the parameters, the roots of which supply the energy eigenvalues. With the = 0 choice the quartic potential term is canceled, leading to the reduced sextic oscillator. It was found that the expressions for the energy eigenvalues and the corresponding wave functions of this potential agree with those obtained from the QES formalism. Possible generalizations of the method are also presented.

Original language | English |
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Journal | Modern Physics Letters A |

DOIs | |

Publication status | Published - 1 Jan 2019 |

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### Keywords

- bi-confluent Heun equation
- quasi-exactly solvable potentials
- Schrödinger equation
- sextic oscillator

### ASJC Scopus subject areas

- Nuclear and High Energy Physics
- Astronomy and Astrophysics

### Cite this

*Modern Physics Letters A*. https://doi.org/10.1142/S0217732319501347

**Exact solutions of the sextic oscillator from the bi-confluent Heun equation.** / Lévai, Géza; Ishkhanyan, Artur M.

Research output: Contribution to journal › Article

*Modern Physics Letters A*. https://doi.org/10.1142/S0217732319501347

}

TY - JOUR

T1 - Exact solutions of the sextic oscillator from the bi-confluent Heun equation

AU - Lévai, Géza

AU - Ishkhanyan, Artur M.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - In this paper, the sextic oscillator is discussed as a potential obtained from the bi-confluent Heun equation after a suitable variable transformation. Following earlier results, the solutions of this differential equation are expressed as a series expansion of Hermite functions with shifted and scaled arguments. The expansion coefficients are obtained from a three-term recurrence relation. It is shown that this construction leads to the known quasi-exactly solvable (QES) form of the sextic oscillator when some parameters are chosen in a specific way. By forcing the termination of the recurrence relation, the Hermite functions turn into Hermite polynomials with shifted arguments, and, at the same time, a polynomial expression is obtained for one of the parameters, the roots of which supply the energy eigenvalues. With the = 0 choice the quartic potential term is canceled, leading to the reduced sextic oscillator. It was found that the expressions for the energy eigenvalues and the corresponding wave functions of this potential agree with those obtained from the QES formalism. Possible generalizations of the method are also presented.

AB - In this paper, the sextic oscillator is discussed as a potential obtained from the bi-confluent Heun equation after a suitable variable transformation. Following earlier results, the solutions of this differential equation are expressed as a series expansion of Hermite functions with shifted and scaled arguments. The expansion coefficients are obtained from a three-term recurrence relation. It is shown that this construction leads to the known quasi-exactly solvable (QES) form of the sextic oscillator when some parameters are chosen in a specific way. By forcing the termination of the recurrence relation, the Hermite functions turn into Hermite polynomials with shifted arguments, and, at the same time, a polynomial expression is obtained for one of the parameters, the roots of which supply the energy eigenvalues. With the = 0 choice the quartic potential term is canceled, leading to the reduced sextic oscillator. It was found that the expressions for the energy eigenvalues and the corresponding wave functions of this potential agree with those obtained from the QES formalism. Possible generalizations of the method are also presented.

KW - bi-confluent Heun equation

KW - quasi-exactly solvable potentials

KW - Schrödinger equation

KW - sextic oscillator

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UR - http://www.scopus.com/inward/citedby.url?scp=85064524346&partnerID=8YFLogxK

U2 - 10.1142/S0217732319501347

DO - 10.1142/S0217732319501347

M3 - Article

AN - SCOPUS:85064524346

JO - Modern Physics Letters A

JF - Modern Physics Letters A

SN - 0217-7323

ER -