Spurious Modes in Dirac Calculations and How to Avoid Them

Chapter
Part of the Mathematical Physics Studies book series (MPST)

Abstract

In this paper we consider the problem of the occurrence of spurious modes when computing the eigenvalues of Dirac operators, with the motivation to describe relativistic electrons in an atom or a molecule. We present recent mathematical results which we illustrate by simple numerical experiments. We also discuss open problems.

Notes

Acknowledgments

M. L. would like to thank Lyonell Boulton and Nabile Boussaid for stimulating discussions, in particular concerning the numerical experiments of this article. M. L. has received financial support from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement MNIQS 258023). M. L. and É. S. acknowledge financial support from the French Ministry of Research (ANR-10-BLAN-0101).

References

  1. 1.
    Aceto, L., Ghelardoni, P., Marletta, M.: Numerical computation of eigenvalues in spectral gaps of Sturm-Liouville operators. J. Comput. Appl. Math. 189, 453–470 (2006)CrossRefMATHMathSciNetADSGoogle Scholar
  2. 2.
    Boffi, D., Brezzi, F., Gastaldi, L.: On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comp. 69, 121–140 (2000)CrossRefMATHMathSciNetADSGoogle Scholar
  3. 3.
    Bossavit, A.: Solving Maxwell equations in a closed cavity, and the question of ‘spurious modes’. IEEE Trans. Magn. 26, 702–705 (1990)CrossRefADSGoogle Scholar
  4. 4.
    Boulton, L., Boussaid, N.: Non-variational computation of the eigenstates of Dirac operators with radially symmetric potentials. LMS J. Comput. Math. 13, 10–32 (2010)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Boulton, L., Boussaid, N., Lewin, M.: Generalised Weyl theorems and spectral pollution in the Galerkin method. J. Spectra. Theor. 2, 329–354 (2012)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Boulton, L., Levitin, M.: On approximation of the eigenvalues of perturbed periodic Schrödinger operators. J. Phys. A 40, 9319–9329 (2007)CrossRefMATHMathSciNetADSGoogle Scholar
  7. 7.
    Bunting, C.F., Davis, W.A.: A functional for dynamic finite-element solutions in electromagnetics. IEEE Trans. Antennas Propag. 47, 149–156 (1999)CrossRefMATHMathSciNetADSGoogle Scholar
  8. 8.
    Cancès, E., Ehrlacher, V., Maday, Y.: Periodic Schrödinger operators with local defects and spectral pollution. SIAM J. Numer. Anal. 50, 3016–3035 (2012)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Csendes, Z.J., Silvester, P.: Numerical solution of dielectric loaded waveguides: I-finite-element analysis. IEEE Trans. Microw. Theor. Tech. 18, 1124–1131 (1970)CrossRefADSGoogle Scholar
  10. 10.
    Davies, E.B.: Spectral enclosures and complex resonances for general self-adjoint operators. LMS J. Comput. Math. 1, 42–74 (1998). (electronic)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Dolbeault, J., Esteban, M.J., Séré, É.: On the eigenvalues of operators with gaps. Application to Dirac operators. J. Funct. Anal. 174, 208–226 (2000)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Drake, G.W.F., Goldman, S.P.: Application of discrete-basis-set methods to the dirac equation. Phys. Rev. A 23, 2093–2098 (1981)CrossRefADSGoogle Scholar
  13. 13.
    Dyall, K.G., Fægri Jr, K.: Kinetic balance and variational bounds failure in the solution of the Dirac equation in a finite Gaussian basis set. Chem. Phys. Lett. 174, 25–32 (1990)CrossRefADSGoogle Scholar
  14. 14.
    Esteban, M.J., Lewin, M., Séré, É.: Variational methods in relativistic quantum mechanics. Bull. Am. Math. Soc. (N.S.). 45, 535–593 (2008)Google Scholar
  15. 15.
    Fernandes, P., Raffetto, M.: Counterexamples to the currently accepted explanation for spurious modes and necessary and sufficient conditions to avoid them. IEEE Trans. Magn. 38, 653–656 (2002)CrossRefADSGoogle Scholar
  16. 16.
    Grant, I.P.: Conditions for convergence of variational solutions of Dirac’s equation in a finite basis. Phys. Rev. A 25, 1230–1232 (1982)CrossRefADSGoogle Scholar
  17. 17.
    Hylleraas, E.A., Undheim, B.: Numerische berechnung der 2S-terme von ortho-und par-helium. Z. Phys. 65, 759–772 (1930)CrossRefMATHADSGoogle Scholar
  18. 18.
    Klahn, B., Bingel, W.A.: The convergence of the Rayleigh-Ritz method in quantum chemistry II. Investigation of the convergence for special systems of slater, gauss and two-electron functions. Theoret. Chim. Acta 44, 27–43 (1977)Google Scholar
  19. 19.
    Kutzelnigg, W.: Basis set expansion of the Dirac operator without variational collapse. Int. J. Quant. Chem. 25, 107–129 (1984)CrossRefGoogle Scholar
  20. 20.
    Lewin, M., Séré, É.: Spectral pollution and how to avoid it (with applications to Dirac and periodic Schrödinger operators). Proc. London Math. Soc. 100, 864–900 (2010)CrossRefMATHGoogle Scholar
  21. 21.
    MacDonald, J.K.L.: Successive approximations by the Rayleigh-Ritz variation method. Phys. Rev. 43, 830–833 (1933)CrossRefADSGoogle Scholar
  22. 22.
    nan Jiang, B., Wu, J., Povinelli, L.: The origin of spurious solutions in computational electromagnetics. J. Comput. Phys. 125, 104–123 (1996)CrossRefMATHMathSciNetADSGoogle Scholar
  23. 23.
    Pestka, G.: Spurious roots in the algebraic Dirac equation. Phys. Scr. 68, 254–258 (2003)CrossRefMATHADSGoogle Scholar
  24. 24.
    Rappaz, J., Sanchez Hubert, J., Sanchez Palencia, E., Vassiliev, D.: On spectral pollution in the finite element approximation of thin elastic “membrane” shells. Numer. Math. 75, 473–500 (1997)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Schroeder, W., Wolff, I.: The origin of spurious modes in numerical solutions of electromagnetic field eigenvalue problems. IEEE Trans. Microw. Theor. Tech. 42, 644–653 (1994)CrossRefADSGoogle Scholar
  26. 26.
    Schwerdtfeger, P.(ed.): Relativistic electronic structure theory. Part 1. Fundamentals of Theoretical and Computational Chemistry, vol. 11. Elsevier (2002) (elsevier ed.)Google Scholar
  27. 27.
    Shabaev, V., Tupitsyn, I.I., Yerokhin, V.A., Plunien, G., Soff, G.: Dual kinetic balance approach to basis-set expansions for the Dirac equation. Phys. Rev. Lett. 93, 130405 (2004)CrossRefADSGoogle Scholar
  28. 28.
    Simon, B.: The theory of schrodinger operators: what’s it all about? Eng. Sci. 48, 20–25 (1985)Google Scholar
  29. 29.
    Stanton, R.E., Havriliak, S.: Kinetic balance: a partial solution to the problem of variational safety in Dirac calculations. J. Chem. Phys. 81, 1910–1918 (1984)CrossRefADSGoogle Scholar
  30. 30.
    Stolz, G., Weidmann, J.: Approximation of isolated eigenvalues of ordinary differential operators. J. Reine Angew. Math. 445, 31–44 (1993)MATHMathSciNetGoogle Scholar
  31. 31.
    Stolz, G., Weidmann, J.: pproximation of isolated eigenvalues of general singular ordinary differential operators. Results Math. 28, 345–358 (1995)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Sutcliffe, B.: What mathematicians know about the solutions of schrodinger coulomb hamiltonian. Should chemists care? J. Math. Chem. 44, 988–1008 (2008)CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    Thaller, B.: The Dirac Equation. Texts and Monographs in Physics, Springer, Berlin (1992)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Département de Mathématiques (CNRS UMR 8088)Université de Cergy-PontoiseCergy-Pontoise CedexFrance
  2. 2.Ceremade (CNRS UMR 7534)Université Paris-Dauphine, Place de Lattre de TassignyParis Cedex 16France

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