A. Galtbayar, A. Jensen, K Yajima
Local time-decay of solutions to Schroedinger equations with time-periodic potentials
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ABSTRACT.  Let $H(t)=-\Delta+V(t,x)$ be a time-dependent 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 quasi-periodic solution.