Asymptotic series for wave functions and energy levels of doubly anharmonic oscillators

Asymptotic series expansions for the wave functions and energy levels of the doubly anharmonic-oscillator system of the $a{x}^2+b{x}^4+c{x}^6$ type have been obtained. The asymptotic expansion for the wave function reduces to a sequence of exact solutions of the Schrödinger equation for special values of certain combinations of the coupling constants. A WKB-type analysis for large values of $n$ (the excitation quantum number) yields an asymptotic expression for the excited energy levels, valid for large values of the dominant coupling. Exact eigenvalues have been computed numerically for a wide range of $n$ and $c$, the dominant coupling. The accuracy of the asymptotic series for the energy eigenvalues of excited states is examined by comparison with the exact eigenvalues obtained numerically and is found to be satisfactory.