02-167 O.Bourget

Singular Continuous Floquet Operator for Systems with Increasing Gaps (307K, Postscript) Apr 4, 02

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Abstract. Consider the Floquet operator of a time independent quantum system, periodically perturbed by a rank one kick, acting on a separable Hilbert space: $e^{-iH_0T}e^{-i\kappa T |\phi \ket \bra \phi|}$ where $T$ and $\kappa$ are the period and the coupling constant respectively. Assume the spectrum of the self adjoint operator $H_0$ is pure point, simple, bounded from below and the gaps between the eigenvalues $(\lambda_n)$ grow like: $\lambda_{n+1} - \lambda_{n} \sim C n^d$ with $d \geq 2$. Under some hypotheses on the arithmetical nature of the eigenvalues and on the vector $\phi$, cyclic for $H_0$, we prove the Floquet operator of the perturbed system has purely singular continuous spectrum.

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