 0469 O.Bourget
 Singular Continuous Floquet Operator for Periodic Quantum Systems
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Mar 8, 04

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Abstract. Consider the Floquet operator of a time independent quantum system, acting on a separable Hilbert space, periodically perturbed by a rank one kick: $e^{iH_0T}e^{i\kappa T \phi\ket\bra\phi}$ where $T$, $\kappa$ are respectively the period and the coupling constant and $H_0$ is a pure point selfadjoint operator, bounded from below. Under some hypotheses on the vector $\phi$, cyclic for $H_0$ we prove the following:
If the gaps between the eigenvalues $(\lambda_n)$ are such that: $\lambda_{n+1}\lambda_{n}\geq C n^{\gamma}$ for some $\gamma \in ]0,1[$ and $C>0$, then the Floquet operator of the perturbed system is purely singular continuous Ta.e.
If $H_0$ is the Hamiltonian of the onedimensional rotator on $L^2({\mathbb R}/T_0{\mathbb Z})$ and the ratio $2\pi T/T_0^2$ is irrational, then the Floquet operator is purely singular continuous as soon as $\kappa T \neq 0(2\pi)$
We also establish an integral formula for the family $(e^{iH_0T}e^{i\kappa T \phi\ket\bra\phi})_{T>0, \kappa \in {\mathbb R}}.$
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