02-497 A. Galtbayar, A. Jensen, K Yajima
Local time-decay of solutions to Schroedinger equations with time-periodic potentials (488K, pdf) Nov 29, 02
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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.

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