Strichartz estimates for higher-order Schr\"odinger equations and their applications

TL;DR

This paper establishes uniform global-in-time Strichartz estimates for higher-order Schr"odinger equations and demonstrates their applications in approximating pseudo-relativistic equations and proving scattering for nonlinear variants.

Contribution

It provides the first uniform Strichartz estimates for higher-order Schr"odinger equations and applies these to analyze pseudo-relativistic and nonlinear equations.

Findings

Uniform global-in-time Strichartz estimates established

Higher-order Hartree-Fock approximates pseudo-relativistic equations accurately

Small data scattering proven for higher-order nonlinear Schr"odinger equations

Abstract

In this paper, we consider the higher-order linear Schr\"odinger equations, that is, a formal finite Taylor expansion of the linear pseudo-relativistic equation. We establish the global-in-time Strichartz estimates for these higher-order equations which hold uniformly in the speed of light. As nonlinear applications, we show that the higher-order Hartree(-Fock) equation approximates the corresponding pseudo-relativistic equation on an arbitrarily long time interval, with higher accuracy than the non-relativistic equation. We also prove small data scattering for the higher-order nonlinear Schr\"odinger equations.