Symmetrized quartic polynomial oscillators and their partial exact solvability

Abstract: Sextic polynomial oscillator is probably the best known quantum system which is partially exactly {\it alias} quasi-exactly solvable (QES), i.e., which possesses closed-form, elementary-function bound states $\psi(x)$ at certain couplings and energies. In contrast, the apparently simpler and phenomenologically more important quartic polynomial oscillator is {\em not\,} QES. A resolution of the paradox is proposed: The one-dimensional Schr\"{o}dinger equation is shown QES after the analyticity-violating symmetrization $V(x)=A|x|+B x^2+C|x|^3+x^4$ of the quartic polynomial potential.