Refined mass-critical Strichartz estimates for Schr\"{o}dinger operators

TL;DR

This paper develops refined Strichartz estimates for time-dependent Schrödinger operators at $L^2$ regularity, crucial for understanding large data solutions to mass-critical nonlinear Schrödinger equations with potentials.

Contribution

It extends bilinear Fourier restriction estimates to nontranslation-invariant Schrödinger operators, advancing harmonic analysis tools for mass-critical NLS with potentials.

Findings

Refined Strichartz estimates at $L^2$ regularity for Schrödinger operators.

Extension of Tao's bilinear restriction estimate to more general operators.

Inverse Strichartz theorem and profile decompositions for large data NLS.

Abstract

We develop refined Strichartz estimates at $L^2$ regularity for a class of time-dependent Schr\"{o}dinger operators. Such refinements begin to characterize the near-optimizers of the Strichartz estimate, and play a pivotal part in the global theory of mass-critical NLS. On one hand, the harmonic analysis is quite subtle in the $L^2$-critical setting due to an enormous group of symmetries, while on the other hand, the spacetime Fourier analysis employed by the existing approaches to the constant-coefficient equation are not adapted to nontranslation-invariant situations, especially with potentials as large as those considered in this article. Using phase space techniques, we reduce to proving certain analogues of (adjoint) bilinear Fourier restriction estimates. Then we extend Tao's bilinear restriction estimate for paraboloids to more general Schr\"{o}dinger operators. As a particular application, the resulting inverse Strichartz theorem and profile decompositions constitute a key harmonic analysis input for studying large data solutions to the $L^2$-critical NLS with a harmonic oscillator potential in dimensions $\ge 2$. This article builds on recent work of Killip, Visan, and the author in one space dimension.