 02497 A. Galtbayar, A. Jensen, K Yajima
 Local timedecay of solutions to Schroedinger equations with timeperiodic potentials
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Nov 29, 02

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Abstract. Let $H(t)=\Delta+V(t,x)$ be a timedependent Schr\"{o}dinger operator on
$L^2(\R^3)$. We assume that $V(t,x)$ is $2\pi$periodic in time and decays
sufficiently rapidly in space. Let $U(t,0)$ be the associated propagator.
For $u_0$ belonging to the continuous spectral subspace of $L^2(\R^3)$ for the
Floquet operator $U(2\pi, 0)$, we
study the behavior of $U(t,0)u_0$ as $t\to\infty$ in the topology of
$x$weighted spaces, in the form of asymptotic expansions. Generically the
leading term is $t^{3/2}B_1u_0$. Here $B_1$ is a finite rank operator mapping
functions of $x$ to functions of $t$ and $x$, periodic in $t$. If
$n\in\Z$ is an eigenvalue, or a threshold resonance of the corresponding
Floquet Hamiltonian $i\pa_t + H(t)$, the leading behavior is
$t^{1/2}B_0u_0$.
The point spectral subspace for $U(2\pi, 0)$ is finite dimensional. If
$U(2\pi, 0)\phi_j = e^{i2\pi\l_j }\phi_j$, then $U(t, 0)\phi_j$ represents
a quasiperiodic solution.
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