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

Abstract

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 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 $\delta =0$ choice the quartic potential term is cancelled, leading to the {\it 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 quasi-exactly solvable formalism. Possible generalizations of the method are also presented.

References in corpus (18)

• Supersymmetry and Quantum Mechanics
• Relativistic quantum dynamics of a charged particle in cosmic string spacetime in the presence of magnetic field and scalar potential
• Quasi-Exactly Solvable Systems and Orthogonal Polynomials
• Exact solution of the Schrödinger equation for the inverse square root potential $V_0/\sqrt{x}$
• The sextic oscillator as a $\gamma$-independent potential
• Solutions of the bi-confluent Heun equation in terms of the Hermite functions
• Schrödinger potentials solvable in terms of the general Heun functions
• The conditionally exactly solvable potentials: A misunderstanding (comment on "Conditionally exactly soluble class of quantum potentials")
• Comprehensive analysis of conditionally exactly solvable models
• Unified treatment of the Coulomb and harmonic oscillator potentials in $D$ dimensions
• Solving a two-electron quantum dot model in terms of polynomial solutions of a Biconfluent Heun Equation
• Discretization of Natanzon potentials
• Analytic solutions of the quantum two-state problem in terms of the double-, bi- and tri-confluent Heun functions
• A conditionally exactly solvable generalization of the inverse square root potential
• The third exactly solvable hypergeometric quantum-mechanical potential
• Potentials of the Heun class
• New conditionally exactly solvable inverse power law potentials
• A singular Lambert-W Schrödinger potential exactly solvable in terms of the confluent hypergeometric functions

Cited by in corpus (3)

• Hermite function solutions of the Schrödinger equation for the sextic oscillator
• Avoided level crossings in quasi-exact approach
• Energy partition for anharmonic, undamped, classical oscillators