Time-dependent Floquet theory and absence of an adiabatic limit

Quantum systems subject to time periodic fields of finite amplitude λ have conventionally been handled either by low-order perturbation theory, for λ not too large, or by exact diagonalization within a finite basis of $N$ states. An adiabatic limit, as λ is switched on arbitrarily slowly, has been assumed. But the validity of these procedures seems questionable in view of the fact that, as $N\to \infty ,$ the quasienergy spectrum becomes dense, and numerical calculations show an increasing number of weakly avoided crossings (related in perturbation theory to high-order resonances). This paper deals with the highly nontrivial behavior of the solutions in this limit. The Floquet states, and the associated quasienergies, become highly irregular functions of the amplitude λ. The mathematical radii of convergence of perturbation theory in λ approach zero. There is no adiabatic limit of the wave functions when λ is turned on arbitrarily slowly. However, the quasienergy becomes independent of $\lambda \mathrm{}\left(t\right)\mathrm{}$ in this limit. We introduce a modification of the adiabatic theorem. We explain why, in spite of the pervasive pathologies of the Floquet states in the limit $N\to \infty ,$ the conventional approaches are appropriate in almost all physically interesting situations.