Arkiv för Matematik

, Volume 37, Issue 1, pp 1–32 | Cite as

Asymptotics of the scattering phase for the Dirac operator: High energy, semi-classical and non-relativistic limits

  • Vincent Bruneau
  • Didier Robert
Article

Abstract

In this paper we prove several results for the scattering phase (spectral shift function) related with perturbations of the electromagnetic field for the Dirac operator in the Euclidean space.

Many accurate results are now available for perturbations of the Schrödinger operator, in the high energy regime or in the semi-classical regime. Here we extend these results to the Dirac operator. There are several technical problems to overcome because the Dirac operator is a system, its symbol is a 4×4 matrix, and its continuous spectrum has positive and negative values. We show that we can separate positive and negative energies to prove high energy asymptotic expansion and we construct a semi-classical Foldy-Wouthuysen transformation in the semi-classical case. We also prove an asymptotic expansion for the scattering phase when the speed of light tends to infinity (non-relativistic limit).

Keywords

Electromagnetic Field Euclidean Space Asymptotic Expansion Accurate Result Technical Problem 

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References

  1. 1.
    Balslev, E. andHelffer, B., Limiting absorption principle and resonances for the Dirac operator,Adv. in Appl. Math. 13 (1992), 186–215.CrossRefMathSciNetGoogle Scholar
  2. 2.
    Birman, M. S. andKrein, M. G., On the theory of wave operators and scattering operators,Dokl. Akad. Nauk. SSSR 144 (1962), 475–478 (Russian). English transl.:Soviet Math. Dokl. 3 (1962), 740–744.MathSciNetGoogle Scholar
  3. 3.
    Brummelhuis, R. andNourrigat, J., Scattering amplitude for Dirac operators, Preprint, Reims, 1997.Google Scholar
  4. 4.
    Bruneau, V., Propriétés asymptotiques du spectre continu d'opérateurs de Dirac, Thesis, Nantes, 1995.Google Scholar
  5. 5.
    Bruneau, V., Sur le spectre continu de l'opérateur de Dirac: formule de Weyl, limite non-relativiste,C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), 43–48.MATHMathSciNetGoogle Scholar
  6. 6.
    Bruneau, V., Fonctions Zeta et Eta en presence de spectre continu,C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 475–480.MATHMathSciNetGoogle Scholar
  7. 7.
    Bruneau, V., Asymptotique de la phase de diffusion à haute énergie pour l'opérateur de Dirac, to appear inAnn. Fac. Sci. Toulouse Math. (1998).Google Scholar
  8. 8.
    Cerbah, S., Principe d'absorption limite semi-classique pour l'opérateur de Dirac, Preprint, Reims, 1995.Google Scholar
  9. 9.
    Chazarain, J., Spectre d'un hamiltonien quantique et mécanique classique,Comm. Partial Differential Equations 5 (1980), 595–644.MATHMathSciNetGoogle Scholar
  10. 10.
    Gérard, C. andMartinez, A., Principe d'absorption limite pour des opérateurs de Schrödinger à longue portée,C. R. Acad. Sci. Paris Sér. I Math. 306 (1989), 121–123.Google Scholar
  11. 11.
    Grigis, A. andMohamed, A., Finitude des lacunes dans le spectre de l'opérateur de Schrödinger et de Diract avec des potentiels électrique et magnétique periodiques,J. Math. Kyoto Univ. 33 (1993), 1071–1096.MathSciNetGoogle Scholar
  12. 12.
    Grigore, D. R., Nenciu, G. andPurice, R., On the nonrelativistic limit of the Dirac Hamiltonian,Ann. Inst. H. Poincaré Phys. Théor. 51, (1989), 231–263.MathSciNetGoogle Scholar
  13. 13.
    Helffer, B. andRobert, D., Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques,Ann. Inst. Fourier (Grenoble) 31:3 (1981), 169–223.MathSciNetGoogle Scholar
  14. 14.
    Helffer, B. andRobert, D., Calcul fonctionnel par la transformation de Melline et opérateurs admissibles,J. Funct. Anal. 53 (1983), 246–268.CrossRefMathSciNetGoogle Scholar
  15. 15.
    Helffer, B. andSjöstrand, J., Analyse semi-classique de l'équation de Harper, II, Comportement semi-classique près d'un rationnel,Mém. Soc. Math. France 40 (1990).Google Scholar
  16. 16.
    Hislop, P. andNakamura, S., Semiclassical resolvent estimates,Ann. Inst. H. Poincaré Phys. Théor. 51 (1989), 187–198.MathSciNetGoogle Scholar
  17. 17.
    Jecko, T., Sections efficaces totales d'une molécule diatomique dans l'approximation de Born-Oppenheimer, Thesis, Nantes, 1996.Google Scholar
  18. 18.
    Jensen, A., Mourre, E. andPerry, P., Multiple commutator estimates and resolvent smoothness in quantum scattering theory,Ann. Inst. H. Poincaré Phys. Théor. 41 (1984), 207–225.MathSciNetGoogle Scholar
  19. 19.
    Robert, D.,Autour de l'approximation semi-classique, Progr. Math.68, Birkhäuser, Boston, Mass., 1987.Google Scholar
  20. 20.
    Robert, D., Asymptotique de la phase de diffusion à haute énergie pour des perturbations du second ordre du Laplacien,Ann. Sci. École Norm. Sup. 25 (1992), 107–134.MATHGoogle Scholar
  21. 21.
    Robert, D., On the scattering theory for long range perturbations of Laplace operators,J. Anal. Math. 59 (1992), 189–203.MATHMathSciNetGoogle Scholar
  22. 22.
    Robert, D., Relative time-delay for perturbations of elliptic operators and semiclassical asymptotics,J. Funct. Anal. 126 (1994), 36–82.CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Robert, D. andTamura, H., Semi-classical asymptotics for local spectral densites and time delay problems in scattering processes,J. Funct. Anal. 80 (1988), 124–147.CrossRefMathSciNetGoogle Scholar
  24. 24.
    Thaller, B.,The Dirac Equation, Texts and Monographs in Phys., Springer-Verlag, Berlin-Heidelberg-New York, 1992.Google Scholar
  25. 25.
    Yajima, K., The quasi-classical approximation to Dirac equation, I,J. Fac. Sci. Univ. Tokyo Sect. I A Math. 29 (1982), 161–194.MATHMathSciNetGoogle Scholar
  26. 26.
    Yamada, O., On the principle of limiting absorption for the Dirac operators,Publ. Res. Inst. Math. Sci. 8 (1972/73), 557–577.MathSciNetGoogle Scholar

Copyright information

© Institut Mittag-Leffler 1999

Authors and Affiliations

  • Vincent Bruneau
    • 1
  • Didier Robert
    • 2
  1. 1.Département de mathématiquesUniversité Bordeaux ITalenceFrance
  2. 2.Département de mathématiquesUniversité de NantesNantes Cedex 03France

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