Strong unique continuation property for the Dirac equation
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, and some lengthy computations. In deriving their results they prove another estimate of the Carleman type: For \(n\geq 2\), let \(r\) denote \(|x|\). Then there exists a positive number \(\gamma_0\) such that for any \(\gamma> \gamma_0\) it is true that \({1\over 2}\|r^{-\gamma-1}u\|_{L2}\leq \|r^{-\gamma}Du\|_{L2}\). The authors also offer a counterexample showing that their uniqueness results are optimal.
scientific article; zbMATH DE number 1496026
scientific article