Quantitative Landis-type result for Dirac operators

TL;DR

This paper establishes sharp quantitative estimates on the decay of solutions to Dirac equations at infinity, using Carleman inequalities and regularity results, with improvements under symmetry and real-valued conditions.

Contribution

It introduces a Landis-type estimate for Dirac operators, providing sharp bounds on solution decay and extending previous results with new techniques.

Findings

Sharp decay bounds of solutions at infinity for Dirac equations

Development of a Carleman inequality for Dirac operators

Improved estimates under symmetry and real-valued assumptions

Abstract

We study quantitative unique continuation at infinity for Dirac equations with bounded matrix-valued potentials. For the massless Dirac operator $\mathcal{D}_n$ in $\mathbb{R}^n$, we establish a Landis-type estimate showing that the vanishing order of any nontrivial bounded solution of $( \mathcal{D}_n + \mathbb{V} ) \varphi = 0$ satisfies a lower bound of order $\exp(-\kappa R^{2} (\log R)^{2})$ as $|x|=R\to \infty$; the quadratic growth in the exponent is sharp, in view of previous known results. Our proof follows a Bourgain--Kenig type approach based on a Carleman inequality for Dirac operators which relies on a local H\"older regularity result, which we also prove. In two dimension, we obtain improved quantitative estimates under symmetry assumptions on the potential $\mathbb{V}$ and for real-valued solutions. Finally, we also derive qualitative Landis-type results for Dirac equations with decaying potentials, including critical decay rates.