Spatial decay and time-asymptotic profiles for solutions of Schrodinger equations

In this paper, we study the relationship between the long time behavior of a solution e(it Delta)phi of the Schrodinger equation on R-N and the asymptotic behavior as vertical bar x vertical bar -> infinity of its Initial value phi. Under appropriate hypotheses on phi we show that, for a fixed 0 < sigma < N, if the sequence of dilations lambda(sigma)(n)phi(lambda(n) (.)) converges in S'(R-N) to psi((.)) as lambda(n) -> infinity, then the rescaled solution t(sigma/2)e(it Delta)phi((.)root t) converges in L-r (R-N), for r sufficiently large, to e(i Delta)psi along the subsequence t(n) = lambda(2)(n). Moreover, we show there n exists an initial value phi (in H-infinity(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(R-N). Finally, we prove analogous results for the nonlinear Schrodinger equation.