Weakly regular Floquet Hamiltonians with pure point spectrum

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.