Floquet Hamiltonians with pure point spectrum

We consider Floquet Hamiltonians of the type $K_F : = - i\partial _t + H_0 + \beta V(\omega t)$ , whereH ₀, a selfadjoint operator acting in a Hilbert space ?, has simple discrete spectrumE ₁<E₂<... obeying a gap condition of the type inf {n ^?α(E _n+1?E_n); n=1, 2,...}>0 for a given α>0,t?V(t) is 2π-periodic andr times strongly continuously differentiable as a bounded operator on ?, ω and β are real parameters and the periodic boundary condition is imposed in time. We show, roughly, that providedr is large enough, β small enough and ω non-resonant, then the spectrum ofK _f is pure point. The method we use relies on a successive application of the adiabatic treatment due to Howland and the KAM-type iteration settled by Bellissard and extended by Combescure. Both tools are revisited, adjusted and at some points slightly simplified.