Converging Perturbative Solutions of the Schrödinger Equation for a Two-Level System with a Hamiltonian Depending Periodically on Time

We study the Schrödinger equation of a class of two-level systems under the action of a periodic time-dependent external field in the situation where the energy difference 2ǫ between the free energy levels is sufficiently small with respect to the strength of the external interaction. Under suitable conditions we show that this equation has a solution in terms of converging power series expansions in ǫ. In contrast to other expansion methods, like in the Dyson expansion, the method we present is not plagued by the presence of “secular terms”. Due to this feature we were able to prove absolute and uniform convergence of the Fourier series involved in the computation of the wave functions and to prove absolute convergence of the ǫ-expansions leading to the “secular frequency” and to the coefficients of the Fourier expansion of the wave function.