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We solve the higher order equations of the theory of the strong perturbationsin quantum mechanics given in M. Frasca, Phys. Rev. A 45, 43 (1992), byassuming that, at the leading order, the wave function goes adiabatically. Thisis accomplished by deriving the unitary operator of adiabatic evolution for theleading order. In this way it is possible to show that at least one of thecauses of the problem of phase-mixing, whose effect is the polynomial increasein time of the perturbation terms normally called secularities, arises from theshifts of the perturbation energy levels due to the unperturbed part of thehamiltonian. An example is given for a two-level system that, anyway, shows asecularity at second order also in the standard theory of small perturbations.The theory is applied to the quantum analog of a classical problem that canbecome chaotic, a particle under the effect of two waves of differentamplitudes, frequencies and wave numbers.Comment: 13 pages, Late

The higher orders of the theory of strong perturbations in quantum mechanics and the secularity problem