Quasi-exactly solvable symmetrized quartic and sextic polynomial oscillators

Abstract

The symmetrized quartic polynomial oscillator is shown to admit an sl(2,$\mathbb{R}$) 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.

14 pages, 4 figures; published version; main changes are in introduction, conclusion, and references

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