Strichartz estimates for Schr\"odinger operators with a non-smooth magnetic potential

TL;DR

This paper establishes Strichartz estimates for Schr"odinger operators with non-smooth, time-independent magnetic and electric potentials that decay polynomially, extending previous results to less regular magnetic fields.

Contribution

It proves Strichartz estimates for Schr"odinger operators with magnetic potentials lacking Sobolev regularity, refining earlier methods that required additional divergence or gradient conditions.

Findings

Strichartz estimates hold for operators with continuous, non-smooth magnetic potentials.

The results apply to potentials with polynomial decay, broadening previous classes.

The work improves upon prior techniques by removing regularity assumptions on the magnetic field.

Abstract

We prove Strichartz estimates for the absolutely continuous evolution of a Schr\"odinger operator $H = (i\nabla + A)^2 + V$ in $\R^n$, $n > 2$. Both the magnetic and electric potentials are time-independent and satisfy pointwise polynomial decay bounds. The vector potential $A(x)$ is assumed to be continuous but need not possess any Sobolev regularity. This work is a refinement of previous methods, which required extra conditions on ${\rm div} A$ or $|\nabla|^{\frac12}A$ in order to place the first order part of the perturbation within a suitable class of pseudo-differential operators.