L^p BOUNDEDNESS OF THE SCATTERING WAVE OPERATORS OF SCHRÖDINGER DYNAMICS WITH TIME-DEPENDENT POTENTIALS AND APPLICATIONS–PART I

This paper establishes the L^p boundedness of wave operators localized at high-frequency for linear Schrödinger equations in R³ with time-dependent potentials. The approach to the proof is based on new cancellation lemmas. As a typical application based on this method, combined with Strichartz estimates is the existence and scattering for nonlinear dispersive equations. For example, we prove global existence and uniform boundedness in L^∞, for a class of Hartree nonlinear Schrödinger equations in L²(R³), allowing the presence of solitons. We also prove the existence of free channel wave operators in L^p(R³) for p > 6.