 96243 P. Duclos, P. Stovicek
 Floquet Hamiltonians with pure point spectrum
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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 $\omega$ nonresonant 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 KAMtype iteration settled by Bellissard and extended
by Combescure. Both tools are revisited,
adjusted and at some points slightly simplified.
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