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

Abstract

This paper establishes the Lp 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 Lp(R3) for p > 6. Show more