Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the strong unique continuation property for the Dirac operator

Biagio Cassano

TL;DR

This work analyzes strong unique continuation for the Dirac operator and identifies the sharp constant governing the origin's behavior in the massless case. It constructs a nontrivial spinor $u$ and a smooth potential $\mathbb V$ on $\mathbb{R}^2\setminus\{0\}$ with $\mathcal D u=\mathbb V u$ and $|\mathbb V(z)|\le(\tfrac12+\varepsilon)/|z|$, proving that the known $C<\tfrac{1}{2}$ threshold is optimal in dimension two. It further investigates continuation properties for stronger singularities $|x|^{-\,\gamma}$ with $\gamma>1$, using a Kelvin transformation to relate origin and infinity cases and constructing explicit sharp examples in $n=2,3$. The paper also establishes strong unique continuation from infinity for $|\mathcal D_n u|\le C|u|/|x|$ with $C<\tfrac{1}{2}$, showing compact support, and demonstrates the existence of noncompact, smooth counterexamples for $C>\tfrac{1}{2}$. Overall, these results sharpen the criticality of UCP and SUCP for the Dirac operator, clarifying the roles of singular potentials and the point-at-infinity regime.

Abstract

In [DO99,KY99], the strong unique continuation property from the origin is established for $H_{loc}^1$-solutions to the massless Dirac differential inequality $|{D}_n u | \leq \frac{C}{|x|}|u|$, in dimension $n\geq 2$ and with $C<\frac12$. We show that $\frac12$ is the largest possibile constant in this result, providing an example in $\mathbb{R}^2$ of a (non-trivial) solution of the inequality. Also, we show properties of unique continuation from the origin for solutions to the inequality $|D_n u | \leq \frac{C}{ |x|^γ}|u|$, for $γ>1$, $C>0$. Finally, we establish the strong unique continuation property for the Dirac operator from the point at infinity.

On the strong unique continuation property for the Dirac operator

TL;DR

This work analyzes strong unique continuation for the Dirac operator and identifies the sharp constant governing the origin's behavior in the massless case. It constructs a nontrivial spinor and a smooth potential on with and , proving that the known threshold is optimal in dimension two. It further investigates continuation properties for stronger singularities with , using a Kelvin transformation to relate origin and infinity cases and constructing explicit sharp examples in . The paper also establishes strong unique continuation from infinity for with , showing compact support, and demonstrates the existence of noncompact, smooth counterexamples for . Overall, these results sharpen the criticality of UCP and SUCP for the Dirac operator, clarifying the roles of singular potentials and the point-at-infinity regime.

Abstract

In [DO99,KY99], the strong unique continuation property from the origin is established for -solutions to the massless Dirac differential inequality , in dimension and with . We show that is the largest possibile constant in this result, providing an example in of a (non-trivial) solution of the inequality. Also, we show properties of unique continuation from the origin for solutions to the inequality , for , . Finally, we establish the strong unique continuation property for the Dirac operator from the point at infinity.
Paper Structure (15 sections, 9 theorems, 69 equations)

This paper contains 15 sections, 9 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Proof of \ref{['thm:main']}
  3. Preliminary definitions
  4. Definition of $u_k$ and $\mathbb{V}_k$ in the annulus $\{\rho_{k+1} \leq \lvert z\rvert\leq\rho_{k}\}$
  5. Definition of the functions $u_k$ and $\mathbb V_k$ in $\{\rho_{k,1} \leq \lvert z\rvert\leq\rho_{k,0}\}$ and $\{\rho_{k,6} \leq \lvert z\rvert\leq\rho_{k,5}\}$
  6. Definition of the functions $u_k$ and $\mathbb V_k$ in $\{\rho_{k,2} \leq \lvert z\rvert\leq\rho_{k,1}\}$
  7. Definition of the functions $u_k$ and $\mathbb V_k$ in $\{\rho_{k,3} \leq \lvert z\rvert\leq\rho_{k,2}\}$
  8. Definition of the functions $u_k$ and $\mathbb V_k$ in $\{\rho_{k,4} \leq \lvert z\rvert\leq\rho_{k,3}\}$
  9. Definition of the functions $u_k$ and $\mathbb V_k$ in $\{\rho_{k,5} \leq \lvert z\rvert\leq\rho_{k,4}\}$
  10. Definition of $u_0$ and $\mathbb{V}_0$ in $\{ \lvert z\rvert\geq \rho_{k_0}\}$
  11. Definition of $u$ and $\mathbb{V}$ and conclusion of the proof
  12. Proof of \ref{['theo:gamma']}, \ref{['theo:gamma.example']} and \ref{['thm:SUCinf']}
  13. Proof of \ref{['theo:gamma']}
  14. Proof of \ref{['theo:gamma.example']}
  15. Proof of \ref{['thm:SUCinf']}

Key Result

Theorem 1.1

Let $n\geq 2$ and $\Omega \subset {\mathbb R}^n$ be a domain containing the origin. Let $u \in H^1_{loc} (\Omega;{\mathbb C}^N)$ be a solution to the differential inequality for and assume that $u$ vanishes at infinite order at the origin, that is Then, $u$ vanishes in $\Omega$.

Theorems & Definitions (15)

  • Theorem 1.1: de1999strongkalf1999note
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Lemma 2.1
  • proof
  • ...and 5 more