Quasi exact solutions for an asymmetric double well potential

Algebraic methods are used to derive approximate quasi exact solutions for a parametric model potential having the functional form V(x) = V₀-Mβx+1/2x²(α+βx)² with M a positive integer and β>0. This yields an asymmetric double well potential provided that the parameters (M, α, β) satisfy the conditions -1/12√3 < ω = -Mβ²/2α³ < 1/12√3. If k = α²/β is sufficiently large there is a high barrier between the two wells, and the quasi exact spectrum is essentially harmonic. More generally, each quasi exact solution is the product of a finite polynomial and a universal asymptotic factor, exp[ -(1/2αx²+1/3βx³)], and is mainly localized in the deeper well, even when the spectrum is significantly non-harmonic. For ω sufficiently small, both the quasi exact spectrum and the lower excited bound state spectrum are determined quite accurately by low order Rayleigh-Schrödinger perturbation theory following a suitable (but non-standard) canonical transformation of the reduced Hamiltonian.