P. Duclos, O. Lev, P. Stovicek, M. Vittot
Weakly regular Floquet Hamiltonians with pure point spectrum
(117K, 35 pages, Latex with AmsArt)
ABSTRACT. We study the Floquet Hamiltonian: -i omega d/dt + H + V(t) as
depending on the parameter omega. We assume that the spectrum of H is
discrete, {h_m (m = 1..infinity)}, with h_m of multiplicity M_m.
and that V is an Hermitian operator, 2pi-periodic in t.
Let J > 0 and set Omega_0 = [8J/9,9J/8]. Suppose that for some sigma > 0:
sum_{m,n such that h_m > h_n} mu_{mn}(h_m - h_n)^(-sigma) < infinity
where mu_{mn} = sqrt(min{M_m,M_n)) M_m M_n. We show that in that case
there exist a suitable norm to measure the regularity of V, denoted
epsilon, and positive constants, epsilon_* & delta_*, such that: if
epsilon < epsilon_* then there exists a measurable subset
|Omega_infinity| > |Omega_0| - delta_* epsilon and the Floquet
Hamiltonian has a pure point spectrum for all omega in Omega_infinity.