, Volume 177, Issue 2, pp 327-347

Floquet Hamiltonians with pure point spectrum

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Abstract

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 ℋ, has simple discrete spectrumE 1<E2<... obeying a gap condition of the type inf {n −α(E n+1−En); 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 ℋ, 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.

Communicated by B. Simon