Multiscale Asymptotic Behavior of the Schrödinger Equation

. In this paper, we construct solutions e it D j of the Schro ¨ dinger equation on R N which have nontrivial asymptotic properties simultaneously on di¤erent time and space scales. More precisely, given m A ð 0 ; N Þ and b b 1 = 2 we consider the set o bm ; r ð j Þ of limit points in L r ð R N Þ as t ! y of t m = 2 ½ e it D j (cid:1)ð(cid:2) t b Þ . We show in particular that, given 0 < n < N and an arbitrary countable set S H ð n ; N Þ , there exists an initial value f such that o bm ; r ð f Þ ¼ L r ð R N Þ for all m A ð 0 ; N Þ and b b 1 = 2 such that m = 2 b A S , and all su‰ciently large r . We also establish a result of a similar nature for a nonlinear Schro¨dinger equation.