On the defect of compactness for the Strichartz estimates of the Schrodinger equations

In this paper, we prove that every sequence of solutions to the linear Schrodinger equation, with bounded data in H-1(R-d), d greater than or equal to 3, can be written, up to a subsequence, as an almost orthogonal sum of sequences of the type h(n)(-(d-2)/2) V((t-t(n))/h(n)(2), (x-x(n))/h(n)), where V is a solution of the linear Schrodinger equation, with a small remainder term in Strichartz norms. Using this decomposition, we prove a similar one for the defocusing H-1-critical nonlinear Schrodinger 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, (C) 2001 Academic Press.