Quasi-exactly solvable symmetrized quartic and sextic polynomial oscillators

TL;DR

This paper demonstrates that certain symmetrized quartic and sextic polynomial oscillators are quasi-exactly solvable, providing explicit solutions and expanding the class of known solvable quantum potentials.

Contribution

It introduces a new symmetrized sextic oscillator and proves its QES nature, extending the set of solvable models with explicit solutions.

Findings

Quartic oscillator admits sl(2,R) algebraization.

Explicit QES solutions for sextic oscillator derived.

New class of QES potentials without known counterparts.

Abstract

The symmetrized quartic polynomial oscillator is shown to admit an sl(2,$\R$) algebraization. Some simple quasi-exactly solvable (QES) solutions are exhibited. A new symmetrized sextic polynomial oscillator is introduced and proved to be QES by explicitly deriving some exact, closed-form solutions by resorting to the functional Bethe ansatz method. Such polynomial oscillators include two categories of QES potentials: the first one containing the well-known analytic sextic potentials as a subset, and the second one of novel potentials with no counterpart in such a class.