P. Duclos, P. Stovicek
Floquet Hamiltonians with pure point spectrum
(66K, LaTeX)
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
$E_10$
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.