Spatial decay and time-asymptotic profiles for solutions of Schrodinger equations Thierry CazenaveFred Weissler 35Q4035Q5542B35Schroedinger groupFourier transformasymptotic behaviordilation propertiesself-similar solutionsscattering theorydynamicl systemschaos In this paper, we study the relationship between the long time behavior of a solution $e^{it\Delta}\varphi$ of the Schroedinger equation on $\mathbb{R}^N$ and the asymptotic behavior as $|x| \to \infty$ of its initial value $\varphi$. Under appropriate hypotheses on $\varphi$ we show that, for a fixed $0 < \sigma < N$, if the sequence of dilations $\lambda_n^{\sigma} \varphi (\lambda_n \cdot)$ converges in $\mathcal{S}'(\mathbb{R}^N)$ to $\psi(\cdot)$ as $\lambda_n \to \infty$, then the rescaled solution $t^{\sigma/2}e^{it\Delta} \varphi (\cdot\sqrt t)$ converges in $L^r(\mathbb{R}^N)$, for $r$ sufficiently large, to $e^{i\Delta}\psi$ along the subsequence $t_n = \lambda_n^2$. Moreover, we show there exists an initial value $\varphi$ (in $H^{\infty} (\mathbb{R}^N)$ if $\sigma > N/2$) such that the set of all possible $\psi$ attainable in this fashion is a closed ball $B$ of an infinite dimensional Banach space. The resulting "universal" solution is therefore asymptotically close along appropriate subsequences to all solutions with initial values in $B$. Furthermore, $e^{i\Delta}$ followed by an appropriate dilation generates a chaotic discrete dynamical system on a compact subset of $L^r(\mathbb{R}^N)$. Finally, we prove analogous results for the nonlinear Schroedinger equation. Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.2664 10.1512/iumj.2006.55.2664 en Indiana Univ. Math. J. 55 (2006) 75 - 118 state-of-the-art mathematics http://iumj.org/access/