zbMATH — the first resource for mathematics

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.
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
Full Text: DOI arXiv
[1] E. De Giorgi and S. Spagnolo, “Sulla convergenza degli integrali dell’energia per operatori ellittici del secondo ordine,” Bollettino della Unione Matematica Italiana, vol. 8, pp. 391-411, 1973. · Zbl 0274.35002
[2] S. Spagnolo, “Sul limite delle soluzioni di problemi di cauchy relativi all’equazione del calore,” Annali della Scuola Normale Superiore di Pisa, vol. 21, pp. 657-699, 1967. · Zbl 0153.42103 · numdam:ASNSP_1967_3_21_4_657_0 · eudml:83441
[3] S. Spagnolo, “Convergence in energy for elliptic operators,” in Proceedings of the 3rd Symposium on the Numerical Solution of Partial Differential Equations, pp. 469-498, Academic Press, College Park, Md, USA, 1976. · Zbl 0347.65034
[4] F. Murat, “H-convergence,” in Séminaire d’Analyse Fonctionelle et Numérique de I’Université d’Alger, 1977.
[5] L. Tartar, Cours Peccot au Collège de France, Paris, France, 1977.
[6] L. Tartar, “Quelques remarques sur I’homogénéisation,” in Proceedings of the Japan-France Seminar on Functional Analysis and Numerical Analysis, pp. 469-482, Japan Society for the Promotion of Science, 1978.
[7] G. Dal Maso, An Introduction to \Gamma -Convergence, Birkhäuser, Boston, Mass, USA, 1993. · Zbl 1138.35301
[8] B. Thaller, The Dirac Equation, Springer, Berlin, Germany, 1993. · Zbl 0765.47023
[9] J. Weidmann, Linear Operators in Hilbert Spaces, Springer, New York, NY, USA, 1980. · Zbl 0434.47001
[10] J. Weidmann, Lineare Operatoren in Hilberträumen, Teubner, Wiesbaden, Germany, 2003. · Zbl 0344.47001
[11] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, Germany, 1976. · Zbl 0342.47009
[12] A. Defranceschi, “An introduction to Homogenization and G-convergence,” in Proceedings of the International Conference on Technology of Plasticity (ICTP ’93), School on Homogenization Lecture Notes, Trieste, Italy, 1993.
[13] N. Svanstedt, “G-convergence of parabolic operators,” Nonlinear Analysis: Theory, Methods & Applications, vol. 36, no. 7, pp. 807-843, 1999. · Zbl 0933.35020 · doi:10.1016/S0362-546X(97)00532-4
[14] M. S. Birman and M. Z. Solomjak, Spectral Theory of Selfadjoint Operators in Hilbert Space, North-Holland, Dordrecht, The Netherlands, 1987. · Zbl 0744.47017
[15] J. Weidmann, “Strong operator convergence and spectral theory of ordinary differential operators,” Universitatis Iagellonicae. Acta Mathematica, no. 34, pp. 153-163, 1997. · Zbl 0964.47004
[16] O. A. Oleinik, A. S. Shamaev, and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Elsevier Science, Amsterdam, The Netherlands, 1992. · Zbl 0768.73003
[17] A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam, The Netherlands, 1978. · Zbl 0404.35001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.