Strichartz estimates for the magnetic Schr\"odinger equation with potentials $V$ of critical decay

TL;DR

This paper establishes Strichartz estimates for the magnetic Schr"odinger equation with potentials of critical decay, extending previous results and analyzing endpoint cases in three dimensions.

Contribution

It extends Strichartz estimates to magnetic Schr"odinger operators with critical decay potentials and investigates endpoint estimates in three dimensions.

Findings

Strichartz estimates hold for admissible pairs under certain conditions.

Extended potential class to Fefferman-Phong class in the electric case.

Identified conditions for equivalence of fractional powers of H and Laplacian in R^3.

Abstract

We study the Strichartz estimates for the magnetic Schr\"odinger equation in dimension $n\geq3$. More specifically, for all Schr\"odinger admissible pairs $(r,q)$, we establish the estimate $$ \|e^{itH}f\|_{L^{q}_{t}(\mathbb{R}; L^{r}_{x}(\mathbb{R}^n))} \leq C_{n,r,q,H} \|f\|_{L^2(\mathbb{R}^n)} $$ when the operator $H= -\Delta_A +V$ satisfies suitable conditions. In the purely electric case $A\equiv0$, we extend the class of potentials $V$ to the Fefferman-Phong class. In doing so, we apply a weighted estimate for the Schr\"odinger equation developed by Ruiz and Vega. Moreover, for the endpoint estimate of the magnetic case in $\mathbb{R}^3$, we investigate an equivalence $$ \| H^{\frac{1}{4}} f \|_{L^r(\mathbb{R}^3)} \approx C_{H,r} \big\| (-\Delta)^{\frac{1}{4}} f \big\|_{L^r(\mathbb{R}^3)} $$ and find sufficient conditions on $H$ and $r$ for which the equivalence holds.