Energy spectrum of the potential V=ax2+x4

Suitable sequences of quasi-exactly solvable Hamiltonians are shown to provide stringent upper bounds to the energy eigenvalues of the bound state potential V = ax2 + x4. Procedures to convert these bounds into even further improved energy estimates are developed. For the quartic anharmonic oscillator (a > 0) case a simple argument is provided to indicate that the conventional small-parameter energy expansion does not converge as a Taylor series. An accurate closed-form parametrization of the entire quartic (a = 0) spectrum is noted. The energy difference between the lowest-lying levels of a quartic double well (a < 0) is satisfactorily recovered and for deep wells a useful expression is deduced for it empirically.