Spectral of quasienergies (Q1353766)

Let \(B\) be the closed unit ball in the space \(C(T^1)^\ast\) of measures on the unit circle \(T^1\). So \(B\) is a complete metric space in the \(w^\ast\)-topology. Let \(\Sigma_{\text{ac}}\) (resp. pp, sc, ps, and pc) be the set of measures in \(B\) which are absolutely continuous (resp. pure point, singular continuous, purely singular, and purely continuous). The fact is used that \(\Sigma_{\text{pc}}\) and \(\Sigma_{\text{ps}}\) are both \(G_\delta\) subsets of \(B\). One shows that if \(\mathcal H\) is a complete metric space and \(F:\mathcal H\to B\) is continuous such that both \(F^{-1}(\Sigma_{\text{pp}})\) and \(F^{-1}(\Sigma_{\text{ac}})\) are dense then \(F^{-1}(\Sigma_{\text{sc}})\) is a dense \(G_\delta\) subset of \(\mathcal H\). This result is applied to the kicked rotor with the time-periodic Hamiltonian \(H(t)=-\alpha d^2/d\theta^2 + v(\theta) \sum_{n=-\infty}^\infty\delta(t-2\pi n)\) in \(L_2(d\theta/2\pi,T^1)\), and to the quasiperiodic Rabi oscillator represented by the operator \(K(\alpha,H)=i\alpha_1\partial/\partial\theta_1+ i\alpha_2\partial/\partial\theta_2 + H(\theta)\) where \(H(\theta)\) is a smooth \(2\times 2\) Hermitian matrix-valued function on the torus \(T^2\). In the first case one proves that (1) the Floquet operator \(U(\alpha,v)\) corresponding to \(H(t)\) is purely continuous for a generic set of parameters \(\alpha\) and \(v\) analytic (generic means dense and \(G_\delta\) in an appropriate complete metric space), and (2) \(U(\alpha,v)\) is purely singular continuous for a generic set of \((\alpha,v)\), with \(v\) of bounded variation. In the second case one proves that \(K(\alpha,H)\) is purely continuous for a generic set of \((\alpha,H)\).