WEAKLY REGULAR FLOQUET HAMILTONIANS WITH PURE POINT SPECTRUM

Abstract

We study the Floquet Hamiltonian -i∂t + H + V(ωt), acting in L2([0,T],ℋ, dt), as depending on the parameter ω = 2π/T. We assume that the spectrum of H in ℋ is discrete, [Formula: see text], but possibly degenerate, and that t ↦ V(t) ∈ ℬ(ℋ) is a 2π-periodic function with values in the space of Hermitian operators on ℋ. Let J > 0 and set [Formula: see text]. Suppose that for some σ > 0 it holds true that ∑hm > hnMmMn (hm - hn)-σ < ∞ where Mm is the multiplicity of hm. We show that in that case there exist a suitable norm to measure the regularity of V, denoted ∊V, and positive constants, ∊⋆ and δ⋆, with the property: if ∊V < ∊⋆ then there exists a measurable subset Ω∞ ⊂ Ω0 such that its Lebesgue measure fulfills |Ω∞| ≥ |Ω0| - δ⋆ ∊V and the Floquet Hamiltonian has a pure point spectrum for all ω∈Ω∞.