Maximal estimates for the Schr\"{o}dinger equation with orthonormal initial data

TL;DR

This paper establishes sharp maximal-in-time and space estimates for the Schrödinger equation with orthonormal initial data, enabling new pointwise convergence results and addressing endpoint problems in Strichartz estimates.

Contribution

It introduces novel maximal estimates for orthonormal systems, extending classical results and solving endpoint issues in the context of the Schrödinger equation.

Findings

Sharp maximal-in-time estimates for orthonormal data

Maximal-in-space estimates addressing endpoint problems

Applications to pointwise convergence for infinitely many fermions

Abstract

For the one-dimensional Schr\"odinger equation, we obtain sharp maximal-in-time and maximal-in-space estimates for systems of orthonormal initial data. The maximal-in-time estimates generalize a classical result of Kenig--Ponce--Vega and allow us obtain pointwise convergence results associated with systems of infinitely many fermions. The maximal-in-space estimates simultaneously address an endpoint problem raised by Frank--Sabin in their work on Strichartz estimates for orthonormal systems of data, and provide a path toward proving our maximal-in-time estimates.