The anharmonic oscillator -d²/dx²+x²+b/x⁴+c/x⁶ for extreme values of the anharmonicity constants

The bx^-4+cx^-6 anharmonic perturbation to the harmonic oscillator is studied in two extreme regimes: large anharmonicity constants, corresponding to b large, c large or both large, and small anharmonicity constants. In the former case, a large order strong coupling expansion in terms of an ad hoc expansion parameter is constructed. In the latter case, the lowest order corrections to the unperturbed energies are obtained. Elementary solutions as well as the consequences of the self-transformation of the Hamiltonian are also analysed.