Quasi-exactly solvable polynomial extensions of the quantum harmonic oscillator

The (analytic) sextic oscillator is often considered as the prototype of quasi-exactly solvable (QES) Schrödinger equations, i.e., those Schrödinger equations for which at some ad hoc couplings a finite number of eigenstates can be found explicitly by algebraic means, while the remaining ones remain unknown. Recently, a (non-analytic) QES symmetrized quartic oscillator was introduced and shown to complete the list of (analytic) QES anharmonic oscillators, which does not contain any quartic one. Here we prove that such a quartic oscillator is amenable to an sl(2, ℝ) description. Furthermore, we generalize it to a symmetrized sextic oscillator. The latter is obtained by parting the real line into two subintervals ℝ⁻ and ℝ⁺ on which the corresponding Schrödinger equations are solved by using the functional Bethe ansatz method, and the resulting wavefunctions and their first derivatives are matched at x = 0. Two categories of QES potentials are obtained: 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. (paper)