Quantitative Landis-type result for Dirac operators

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 $\exp(-κR^{2} (\log R)^{2})$ 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 Hölder 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.

Theorems & Definitions (22)

• Theorem 1.1
• Definition 2.1: Weak Solution
• Remark 2.2
• Proposition 2.3
• Proposition 2.4
• Theorem 3.1: Local Hölder regularity
• proof : Proof of Theorem \ref{['thm:holder']}
• Lemma 3.2: Global $L^p$ estimate for compactly supported spinors
• proof
• Remark 3.3