Lower Bounds for Order of Decay or of Growthin Time for Solutions to Linear and Non-linear Schrödinger Equations

We Study lower bounds of decay (or of growth) order in time for solutions to the Cauchy problem for the Schrodinger equation: where f is a linear or non-linear complex-valued function. Under some conditions on f and φ, it is shown that every nontrivial solution u has the estimate for sufficiently large k>0 and for any q∈[2, ∞]. In the previous work [12] of the first named author, we imposed on the assumption that u is asymptotically free. In this article, however, we shall show the assumption is, in fact, irrelevant to the results.