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Non-variational computation of the eigenstates of Dirac operators with radially symmetric potentials

Part of: Quantum theory Special classes of linear operators

Published online by Cambridge University Press:  01 January 2010

Lyonell Boulton  and
Nabile Boussaid
Lyonell Boulton
Affiliation:
Ceremade (UMR CNRS 7534) Université Paris-Dauphine, Place de Lattre de Tassigny, F-75775 Paris Cedex 16, France Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot-Watt University, Edinburgh, EH14 4AS, United Kingdom (email: L.Boulton@hw.ac.uk)
Nabile Boussaid
Affiliation:
Département de Mathématiques, UFR Sciences et techniques, 16 route de Gray - 25 030, Besançon Cedex, France (email: nabile.boussaid@univ-fcomte.fr)

Abstract

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We discuss a novel strategy for computing the eigenvalues and eigenfunctions of the relativistic Dirac operator with a radially symmetric potential. The virtues of this strategy lie in the fact that it avoids completely the phenomenon of spectral pollution and it always provides two-sided estimates for the eigenvalues with explicit error bounds on both eigenvalues and eigenfunctions. We also discuss convergence rates of the method and illustrate our results with various numerical experiments.

MSC classification

Secondary: 47B36: Jacobi (tridiagonal) operators (matrices) and generalizations 47B39: Difference operators 81-08: Computational methods
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 10 - 32
Copyright
Copyright © London Mathematical Society 2010

References

[1] Berthier, A. and Georgescu, V., ‘On the point spectrum of Dirac operators’, J. Funct. Anal. 71 (1987) 309–338.CrossRefGoogle Scholar
[2] Betcke, T., Higham, N., Mehrmann, V., Schröder, C. and Tisseur, F., NLEVP: A collection of nonlinear eigenvalue problems. http://www.mims.manchester.ac.uk/research/numerical-analysis/nlevp.html.Google Scholar
[3] Betcke, T., Higham, N., Mehrmann, V., Schröder, C. and Tisseur, F., NLEVP: A collection of nonlinear eigenvalue problems. MIMS EPrint 2008.40, Manchester Institute for Mathematical Sciences, The University of Manchester, UK, (April 2008).Google Scholar
[4] Birman, M. and Laptev, A., ‘Discrete spectrum of the perturbed Dirac operator’, Ark. Mat. 32 (1994) 13–32.CrossRefGoogle Scholar
[5] Boulton, L., ‘Limiting set of second order spectra’, Math. Comput. 75 (2006) 1367–1382.CrossRefGoogle Scholar
[6] Boulton, L., ‘Non-variational approximation of discrete eigenvalues of self-adjoint operators’, IMA J. Numer. Anal. 27 (2007) 102–121.CrossRefGoogle Scholar
[7] Boulton, L. and Levitin, M., ‘On approximation of the eigenvalues of perturbed periodic Schrodinger operators’, J. Phys. A: Math. Theor. 40 (2007) 9319–9329.CrossRefGoogle Scholar
[8] Dyall, K., ‘Kinetic balance and variational bounds failure in the solution of the Dirac equation in a finite gaussian basis set’, Chem. Phys. Lett. 174 (1990) 25–32.CrossRefGoogle Scholar
[9] Esteban, M., Lewin, M. and Séré, E., ‘Variational methods in relativistic quantum mechanics’, Bull. AMS 45 (2008) 535–593.CrossRefGoogle Scholar
[10] Griesemer, M. and Lutgen, J., ‘Accumulation of discrete eigenvalues of the radial Dirac operator’, J. Funct. Anal. 162 (1999) 120–134.CrossRefGoogle Scholar
[11] Hislop, P. D., ‘Exponential decay of two-body eigenfunctions: a review’, Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory (Berkeley, CA, 1999), Electron. J. Differ. Equ. Conf., vol. 4 (Southwest Texas State University, San Marcos, TX, 2000) 265–288 (electronic).Google Scholar
[12] Klaus, M., ‘On the point spectrum of Dirac operators’, Helv. Phys. Acta. 53 (1980) 463–482.Google Scholar
[13] Langer, H., Langer, M. and Tretter, C., ‘Variational principles for eigenvalues of block operator matrices’, Indiana Univ. Math. J. 51 (2002) 1427–1459.CrossRefGoogle Scholar
[14] Levitin, M. and Shargorodsky, E., ‘Spectral pollution and second-order relative spectra for self-adjoint operators’, IMA J. Numer. Anal. 24 (2004) 393–416.CrossRefGoogle Scholar
[15] Schmidt, K. M., ‘Eigenvalue asymptotics of perturbed periodic Dirac systems in the slow-decay limit’, Proc. Amer. Math. Soc. 131 (2003) 1205–1214 (electronic).CrossRefGoogle Scholar
[16] Shargorodsky, E., ‘Geometry of higher order relative spectra and projection methods’, J. Operator Theory 44 (2000) 43–62.Google Scholar
[17] Stanton, R. and Havriliak, S., ‘Kinetic balance a partial solution to the problem of variational safety in dirac calculations’, J. Chem. Phys. 81 (1984) 1910–1918.CrossRefGoogle Scholar
[18] Strauss, M., ‘Quadratic projection methods for approximating the spectrum of self-adjoint operators’, IMA J. Numer. Anal., to appear (2010).CrossRefGoogle Scholar
[19] Thaller, B., The Dirac Equation (Springer, Berlin, 1992).CrossRefGoogle Scholar
[20] Triebel, H., ‘Hardy inequalities in function spaces’, Math. Bohem. 124 (1999) 123–130.CrossRefGoogle Scholar