Strichartz estimates for the Schr\"odinger equation in high dimensional critical electromagnetic fields

TL;DR

This paper establishes Strichartz estimates for the Schrödinger equation with critical electromagnetic potentials in high dimensions, introducing new kernel construction techniques to handle Coulomb-like fields.

Contribution

It provides the first proof of Strichartz estimates for Schrödinger equations with scaling-critical electromagnetic potentials, including Coulomb potentials, in dimensions n ≥ 3.

Findings

Proves $L^1 o L^ty$ bounds for localized Schrd6dinger propagator.

Establishes global Strichartz estimates in high dimensions.

Introduces novel kernel construction methods for critical electromagnetic fields.

Abstract

We prove Strichartz estimates for the Schr\"odinger equation with scaling-critical electromagnetic potentials in dimensions $n\geq3$. The decay assumption on the magnetic potentials is critical, including the case of the Coulomb potential. Our approach introduces novel techniques, notably the construction of Schwartz kernels for the localized Schr\"odinger propagator, which separates the antipodal points of $\mathbb{S}^{n-1}$, in these scaling critical electromagnetic fields. This method enables us to prove the $L^1(\mathbb{R}^n)\to L^\infty(\mathbb{R}^n)$ for the localized Schr\"odinger propagator, as well as global Strichartz estimates. Our results provide a positive answer to the open problem posed in arXiv:0901.4024 arXiv:1611.04805 arXiv:0806.0778, and fill a longstanding gap left by arXiv:arXiv:0705.0546 arXiv:archive/0608699.