Exact solutions of the sextic oscillator from the biconfluent Heun equation
Abstract
In this paper, the sextic oscillator is discussed as a potential obtained from the biconfluent 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 threeterm recurrence relation. It is shown that this construction leads to the known quasiexactly 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 δ = 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.
 Publication:

Modern Physics Letters A
 Pub Date:
 June 2019
 DOI:
 10.1142/S0217732319501347
 arXiv:
 arXiv:1904.09488
 Bibcode:
 2019MPLA...3450134L
 Keywords:

 Schrödinger equation;
 sextic oscillator;
 biconfluent Heun equation;
 quasiexactly solvable potentials;
 03.65.Ge;
 02.30.Gp;
 02.30.Hq;
 02.30.Ik;
 Solutions of wave equations: bound states;
 Special functions;
 Ordinary differential equations;
 Integrable systems;
 Quantum Physics
 EPrint:
 Modern Phys. Lett. A 34, 1950134 (2019)