Spectral pollution and how to avoid it

Proceedings of the London Mathematical Society

Volumn 100, Issue 3, 2010, Pages 864-900

  Lewin, Mathieu ^a   Séré, Éric ^b  

^a UNIVERSITÉ DE CERGY PONTOISE   (France)

^b UNIVERSITÉ PARIS DAUPHINE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77951785759     PISSN: 00246115     EISSN: 1460244X     Source Type: Journal    
DOI: 10.1112/plms/pdp046     Document Type: Article

References (41)

1. L. ACETO, P. GHELARDONI and M. MARLETTA, 'Numerical computation of eigenvalues in spectral gaps of Sturm-Liouville operators', J. Comput. Appl. Math. 189 (2006) 453-470.
2. L. BOULTON, 'Non-variational approximation of discrete eigenvalues of self-adjoint operator', IMA J. Numer. Anal. 27 (2007) 102-121.
3. L. BOULTON and N. BOUSSAID, 'Non-variational computation of the eigenstates of Dirac operators with radially symmetric potentials', LMS J. Comput. Math. to appear.
4. L. BOULTON and M. LEVITIN, 'On approximation of the eigenvalues of perturbed periodic Schrödinger operators', J. Phys. A 40 (2007) 9319-9329.
5. C. BROUDER, G. PANATI, M. CALANDRA, C. MOUROUGANE and N. MARZARI, 'Exponential localization of Wannier functions in insulators', Phys. Rev. Lett., 98 (2007) 046402.
6. É. CANCÈS, A. DELEURENCE and M. LEWIN, 'Non-perturbative embedding of local defects in crystalline materials', J. Phys.: Condens. Matter 20 (2008) 294213.
7. É. CANCÉS, A. DELEURENCE and M. LEWIN, 'A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case', Commun. Math. Phys. 281 (2008) 129-177.
8. I. CATTO, C. LE BRIS and P.-L. LIONS, 'On the thermodynamic limit for Hartree-Fock type models', Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001) 687-760.
9. E. DAVIES, Spectral theory and differential operators, Cambridge Studies in Advanced Mathematics 42 (Cambridge University Press, Cambridge, 1995).
10. E. DAVIES and M. PLUM, 'Spectral pollution', IMA J. Numer. Anal. 24 (2004) 417-438.
11. J. DESCLOUX, 'Essential numerical range of an operator with respect to a coercive form and the approximation of its spectrum by the galerkin method', SIAM J. Numer. Anal. 18 (1981) 1128-1133.
12. J. DOLBEAULT, M. J. ESTEBAN and E. SÉRÉ, 'On the eigenvalues of operators with gaps. Application to Dirac operators', J. Funct. Anal. 174 (2000) 208-226.
13. J. DOLBEAULT, M. J. ESTEBAN, M. LOSS and L. VEGA, 'An analytical proof of Hardy-like inequalities related to the Dirac operator', J. Funct. Anal. 216 (2004) 1-21.
14. G. W. F. DRAKE and S. P. GOLDMAN, 'Application of discrete-basis-set methods to the Dirac equation', Phys. Rev. A 23 (1981) 2093-2098.
15. K. G. DYALL and K. FæGRI JR, 'Kinetic balance and variational bounds failure in the solution of the Dirac equation in a finite Gaussian basis set', Chem. Phys. Letters 174 (1990) 25-32.
16. M. J. ESTEBAN and M. LOSS, 'Self-adjointness for Dirac operators via Hardy-Dirac inequalities', J. Math. Phys. 48 (2007) 112107, 8.
17. M. J. ESTEBAN, M. LEWIN and É. SÉRÉ, 'Variational methods in relativistic quantum mechanics', Bull. Amer. Math. Soc. (N.S.) 45 (2008) 535-593.
18. W. D. EVANS, P. PERRY and H. SIEDENTOP, 'The spectrum of relativistic one-electron atoms according to Bethe and Salpeter', Comm. Math. Phys. 178 (1996) 733-746.
19. I. P. GRANT, 'Conditions for convergence of variational solutions of Dirac's equation in a finite basis', Phys. Rev. A 25 (1982) 1230-1232.
20. C. HAINZL, M. LEWIN and É. SÉRÉ, 'Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation', Comm. Math. Phys., 257 (2005) 515-562.
21. C. HAINZL, M. LEWIN and E. SÉRÉ, 'Existence of atoms and molecules in the mean-field approximation of no-photon quantum electrodynamics', Arch. Ratio. Mech. Anal. 192 (2009) 453-499.
22. A. C. HANSEN, 'On the approximation of spectra of linear operators on Hilbert spaces', J. Funct. Anal. 254 (2008) 2092-2126.
23. W. KUTZELNIGG, 'Basis set expansion of the Dirac operator without variational collapse', Int. J. Quantum Chemistry, 25 (1984) 107-129.
24. M. LEVITIN and E. SHARGORODSKY, 'Spectral pollution and second-order relative spectra for self-adjoint operators', IMA J. Numer. Anal. 24 (2004) 393-416.
25. N. MARZARI and D. VANDERBILT, 'Maximally localized generalized Wannier functions for composite energy bands', Phys. Rev. B 56 (1997) 12847-12865.
26. G. NENCIU, 'Existence of the exponentially localised Wannier functions', Comm. Math. Phys. 91 (1983) 81-85.
27. G. PANATI, 'Triviality of Bloch and Bloch-Dirac bundles', Ann. Henri Poincare 8 (2007) 995-1011.
28. G. PESTKA, 'Spurious roots in the algebraic Dirac equation', Phys. Scr. 68 (2003) 254-258.
29. A. POKRZYWA, 'Method of orthogonal projections and approximation of the spectrum of a bounded operator', Studia Math. 65 (1979) 21-29.
30. A. POKRZYWA, 'Method of orthogonal projections and approximation of the spectrum of a bounded operator II', Studia Math. 70 (1981) 1-9.
31. M. REED and B. SIMON, 'Methods of modern mathematical physics IV. Analysis of operators (Academic Press, New York, 1978).
32. E. SEILER and B. SIMON, 'Bounds in the Yukawa2 quantum field theory: upper bound on the pressure, Hamiltonian bound and linear lower bound', Comm. Math. Phys. 45 (1975) 99-114.
33. V. SHABAEV, I. I. TUPITSYN, V. A. YEROKHIN, G. PLUNIEN and G. SOFF, 'Dual kinetic balance approach to basis-set expansions for the Dirac equation', Phys. Rev. Lett. 93 (2004) 130405.
34. B. SIMON, 'Trace ideals and their applications, London Mathematical Society Lecture Note Series 35 (Cambridge University Press, Cambridge, 1979).
35. R. E. STANTON and S. HAVRILIAK, 'Kinetic balance: A partial solution to the problem of variational safety in Dirac calculations', J. Chem. Phys. 81 (1984) 1910-1918.
36. G. STOLZ and J. WEIDMANN, 'Approximation of isolated eigenvalues of ordinary differential operators', J. Reine Angew. Math., 445 (1993) 31-44.
37. G. STOLZ and J. WEIDMANN, 'Approximation of isolated eigenvalues of general singular ordinary differential operators', Results Math. 28 (1995) 345-358.
38. J. D. TALMAN, 'Minimax principle for the Dirac equation', Phys. Rev. Lett. 57 (1986) 1091-1094.
39. B. THALLER, The Dirac equation, Texts and Monographs in Physics (Springer-Verlag, Berlin, 1992).
40. L. E. THOMAS, 'Time dependent approach to scattering from impurities in a crystal', Comm. Math. Phys. 33 (1973) 335-343.
41. G. H. WANNIER, 'The structure of electronic excitation levels in insulating crystals', Phys. Rev. 52 (1937) 191-197.