Weakly Regular Floquet Hamiltonians with Pure Point Spectrum

We study the Floquet Hamiltonian −i∂t + H + V (ωt), acting in L([ 0, T ],H, dt), as depending on the parameter ω = 2π/T . We assume that the spectrum of H in H is discrete, Spec(H) = {hm}m=1, but possibly degenerate, and that t 7→ V (t) ∈ B(H) is a 2π-periodic function with values in the space of Hermitian operators on H. Let J > 0 and set Ω0 = [ 89J, 98J ]. Suppose that for some σ > 0 it holds true that ∑ hm>hn μmn(hm − hn) < ∞ where μmn = (min{Mm,Mn})MmMn and Mm is the multiplicity of hm. We show that in that case there exist a suitable norm to measure the regularity of V , denoted ǫV , and positive constants, ǫ⋆ and δ⋆, with the property: if ǫV < ǫ⋆ then there exists a measurable subset Ω∞ ⊂ Ω0 such that its Lebesgue measure fulfills |Ω∞| ≥ |Ω0| − δ⋆ǫV and the Floquet Hamiltonian has a pure point spectrum for all ω ∈ Ω∞.