Strong unique continuation property for the Dirac equation
type: Item
instance of: scholarly article
title: Strong unique continuation property for the Dirac equation
zbMATH Open document ID: 0960.35013
author: Laura De Carli
author: Takashi Ōkaji
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2001-05-10T00:00:00Z
review text: The main purpose of this paper is to prove unique continuation for Dirac operators with potentials dominated by a Coulomb type singularity. Basically, the Dirac operator \(D\) can be regarded as a square root of the positive \(-\Delta I\) operator, where \(\Delta\) is the Laplacian. Let \(\alpha_j\), \(j= 1,2,3\) be \(4\times 4\) matrices obeying the relations \(\alpha^*_j= \alpha_j,\alpha^2_j= I\), and for \(j\neq k\), \(\alpha_j \alpha_k+ \alpha_k \alpha_j= 0\), \(\alpha_0= \left(\begin{smallmatrix} I\\ & -I\end{smallmatrix}\right)\). Let \(\sigma_1\), \(\sigma_2\), \(\sigma_3\) be the \(2\times 2\) Pauli matrices and \(\alpha_j= \left(\begin{smallmatrix} &\sigma j\\ \sigma j\end{smallmatrix}\right)\), with \(j= 1,2,3\). Then the Dirac operator is defined by the formula: \(D= -i\sum^3_{j=1} \alpha_j\partial_{xj}\). The authors prove the following theorems: Let \(W(x)= w(|x|)I\), with \(|x|w(|x|)\in L^\infty(\mathbb{R}^n)\), with \(n\geq 2\). Let \(\Omega\subset \mathbb{R}^n\) be a domain containing the origin, and \(V(x)\) satisfy conditions similar to \(W(x)\), and assume that \(\sup|x|\|V(x)- W(x)\|<{1\over 2}\). Then if \(u\in \{W^{1,2}_{\text{loc}}(\Omega)^N\}\) satisfies the equation \(Du+ Vu= 0\) in \(\Omega\) and vanishes at infinite order at the origin, then \(u\) is identically zero. With similar conditions assigned to \(u\) they also prove that, if \(u\) obeys the inequality \(|Du(x)|\leq C|u(x)|/|x|\) for \(x\in\Omega\), with constant \(C<{1\over 2}\), then \(u\) is identically equal to zero if it assumes the zero value of infinite order at some point in \(\Omega\). The authors make some insightful comments on many results scattered in the literature concerning uniqueness of solutions for the Dirac operator. Their proofs use Carleman's estimates of the type: \[ \int^{+\infty}_{-\infty} e^{-2\gamma y}|\partial_yf+\mu f|^2 dy\geq \int^{+\infty}_{-\infty} |(\partial_y+ \gamma+\mu)v|^2 dy, \] where \(v= e^{-2\gamma y}f\), decomposition into Fourier series, …
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 35B60
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35J45
Mathematics Subject Classification ID: 35R45
zbMATH DE Number: 1496026
zbMATH Keywords: relativistic notions of particles
zbMATH Keywords: Coulomb type singularity
zbMATH Keywords: Carleman's estimates
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195143357
OpenAlex ID: W1969970304
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DOI: 10.2977/PRIMS/1195143357
description: scientific article; zbMATH DE number 1496026