LOCATION: Maryam Mirzakhani Seminar Room at BCAM

Abstract
The Strichartz estimate is one of fundamental tools in the study of nonlinear dispersive equations, especially scattering theory. This talk deals with (global-in-time) Strichartz estimates Schrödinger equations with potentials decaying at infinity. The case when the potential decays sufficiently fast has been extensively studied in the last three decades. However, it has remained mostly unknown for slowly decaying potentials in which case the standard argument, based on the local smoothing effects for the perturbed equation and the free Strichartz estimate, does not work. We instead employ several techniques from long-range scattering theory and microlocal/semiclassical analysis, and prove Strichartz estimates for a class of positive potentials decaying arbitrarily slowly. A typical example is the positive Coulomb potential in three space dimensions. As an application, we also obtain a modified scattering type result for the final state problem of the nonlinear Schrödinger equations with long-range nonlinearity and potential. This is partly joint work with Masaki Kawamoto (Ehime University).

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