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

Géza Lévai, Artur M. Ishkhanyan

Research output: Contribution to journalArticle

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 languageEnglish
JournalModern Physics Letters A
DOIs
Publication statusPublished - 1 Jan 2019

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oscillators
polynomials
eigenvalues
series expansion
differential equations
wave functions
formalism
expansion
energy
coefficients

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

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

In: Modern Physics Letters A, 01.01.2019.

Research output: Contribution to journalArticle

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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.

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