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G-convergence of Dirac operators. (English) Zbl 1259.47018
Let \(\mathcal Y\) be a Hilbert space and let \(\lambda \geq 0\). Denote by \(\mathcal D_\lambda (\mathcal Y)\) the set of not necessarily densely defined operators \(A\) in \(\mathcal Y\) such that \(A\geq \lambda \) and \(A\) is self-adjoint in \(\overline{{\mathbf D}(A)}\). A sequence \((A_h)\) in \(\mathcal D_\lambda (\mathcal Y)\) is said to \(G\)-converge to \(A\) in \(\mathcal D_\lambda (\mathcal Y)\) if \(A_h^{-1}P_h\to A^{-1}P\) (in the strong or weak sense), where \(P\) is the orthogonal projection onto \({\mathbf D}(A)\). The authors show that \(G\)-convergence implies (weak) convergence of corresponding eigenvalues and eigenvectors. This result is applied to perturbations of Dirac operators, restricted to the spectral subspaces associated with the eigenvalues in the gap of the continuous spectrum.
MSC:
47A58 Linear operator approximation theory
47A55 Perturbation theory of linear operators
35Q41 Time-dependent Schrödinger equations and Dirac equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35P05 General topics in linear spectral theory for PDEs
49J45 Methods involving semicontinuity and convergence; relaxation
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