96-243 P. Duclos, P. Stovicek

Floquet Hamiltonians with pure point spectrum (66K, LaTeX) Jun 4, 96

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Abstract. We consider Floquet Hamiltonians of the type $K_F:=-i\partial_t+H_0+\beta V(\omega t)$ where $H_0$, a selfadjoint operator acting in a Hilbert space ${\cal H}$, has simple discrete spectrum $E_1<E_2<\dots$ obeying a gap condition of the type $\inf\{n^{-\alpha}(E_{n+1}-E_n);\ n=1,2,...\}>0$ for a given $\alpha>0$, $t\mapsto V(t)$ is $2\pi$-periodic and $r$ times strongly continuously differentiable as a bounded operator on ${\cal H}$, $\omega$ and $\beta$ are real parameters and the periodic boundary condition is imposed in time. We show, roughly, that provided $r$ is large enough, $\beta$ small enough and $\omega$ non-resonant then the spectrum of $K_F$ is pure point. The method we use relies on a successive application of the adiabatic treatment due to Howland and the KAM-type iteration settled by Bellissard and extended by Combescure. Both tools are revisited, adjusted and at some points slightly simplified.

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