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\title{Dispersive properties of Schr\"odinger equations
 with potentials periodic in time}

\author{Kenji Yajima}

\date{Tokyo and Munich}

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\begin{document}
\maketitle

Let $H(t)= -\Delta+V(t,x)$ be Schr\"odinger operators with time
periodic potentials on $L^2(\mathbb{R}^3)$ with time periodic
potentials $V(t,x)$ with period $2\pi$ which decay sufficiently
rapidly as $|x|\to\infty$. Let $U(t,0)$ be the associated
propagator. We show that the eigenvalues of the Floquet operator
$U(2\pi,0)$ are finite in number and with finite multiplicities
and that, for $u_0$ belonging to the continuous spectral subspace
for $U(2\pi,0)$, the solution $U(t,0)u_0$ satisfies various
dispersive properties: If $u_0$ decays rapidly as $|x|\to\infty$,
$U(t,0)u_0$ has an asymptotic expansion in $t$ as $t\to\infty$ in
the topology of $x$-weighted $L^2$ spaces; it satisfies the
Strichartz estimate and the local smoothing property globally in
time. We use the extended phase space formalism and the Floquet
Hamiltonian for proving these properties.
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