Strichartz and Smoothing Estimates for Schr\"odinger Operators with Almost Critical Magnetic Potentials in Three and Higher Dimensions

TL;DR

This paper establishes smoothing and Strichartz estimates for magnetic Schr"odinger operators with almost critical potentials in higher dimensions, using novel angular decomposition and oscillatory integral techniques.

Contribution

It introduces new methods to prove estimates for magnetic Schr"odinger operators under near-optimal decay and regularity conditions, especially for large gradient perturbations.

Findings

Proved smoothing and Strichartz estimates under minimal potential conditions.

Established a limiting absorption principle for large gradient perturbations.

Developed a uniform angular decomposition approach for resolvent estimates.

Abstract

In this paper we consider magnetic Schr\"odinger operators in R^n, n \ge 3. Under almost optimal conditions on the potentials in terms of decay and regularity we prove smoothing and Strichartz estimates, as well as a limiting absorption principle. For large gradient perturbations the latter is not a corollary of the free case as the differentiated free resolvent does not have small operator norm on any weighted L^2 spaces. We instead show that the spectral radius of such operators decreases to zero, hence their perturbation of the identity is still invertible. The key estimates are based on an angular decomposition of the free resolvent, or rather a bound that holds uniformly for all possible angular decompositions. The proof avoids the Fourier transform and instead uses H\"ormander's variable coefficient Plancherel theorem for oscillatory integrals.