Symmetrized quartic polynomial oscillators and their partial exact solvability

• Quasi-exact solution (QES) method generalized to cover non-analytic potentials.
• The necessary matching of wave function found facilitated in QES framework.
• Sample construction provided for symmetrized quartic oscillators.
• Phenomenological appeal of double-well shape of potential emphasized.

Sextic polynomial oscillator is probably the best known quantum system which is partially exactly alias   quasi-exactly solvable (QES), i.e., which possesses closed-form, elementary-function bound states ψ(x)ψ(x) at certain couplings and energies. In contrast, the apparently simpler and phenomenologically more important quartic polynomial oscillator is not   QES. A resolution of the paradox is proposed: The one-dimensional Schrödinger equation is shown QES after the analyticity-violating symmetrization V(x)=A|x|+Bx2+C|x|3+x4V(x)=A|x|+Bx2+C|x|3+x4 of the quartic polynomial potential.