On the Defect of Compactness for the Strichartz Estimates of the Schrödinger Equations

In this paper, we prove that every sequence of solutions to the linear Schrödinger equation, with bounded data in H1( d), d 3, can be written, up to a subsequence, as an almost orthogonal sum of sequences of the type h−(d−2)/2nV((t−tn)/h2n, (x−xn)/ hn ), where V is a solution of the linear Schrödinger equation, with a small remainder term in Strichartz norms. Using this decomposition, we prove a similar one for the defocusing H1-critical nonlinear Schrödinger equation, assuming that the initial data belong to a ball in the energy space where the equation is solvable. This implies, in particular, the existence of an a priori estimate of the Strichartz norms in terms of the energy.