Skip to main content
SpringerLink
Log in

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

  • Published:
Arkiv för Matematik

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).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Balslev, E. andHelffer, B., Limiting absorption principle and resonances for the Dirac operator,Adv. in Appl. Math. 13 (1992), 186–215.

    Article  MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  3. Brummelhuis, R. andNourrigat, J., Scattering amplitude for Dirac operators, Preprint, Reims, 1997.

  4. Bruneau, V., Propriétés asymptotiques du spectre continu d'opérateurs de Dirac, Thesis, Nantes, 1995.

  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.

    MATH  MathSciNet  Google Scholar 

  6. Bruneau, V., Fonctions Zeta et Eta en presence de spectre continu,C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 475–480.

    MATH  MathSciNet  Google Scholar 

  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).

  8. Cerbah, S., Principe d'absorption limite semi-classique pour l'opérateur de Dirac, Preprint, Reims, 1995.

  9. Chazarain, J., Spectre d'un hamiltonien quantique et mécanique classique,Comm. Partial Differential Equations 5 (1980), 595–644.

    MATH  MathSciNet  Google Scholar 

  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. 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.

    MathSciNet  Google Scholar 

  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.

    MathSciNet  Google Scholar 

  13. Helffer, B. andRobert, D., Comportement semi-classique du spectre des hamiltoniens quantiques elliptiques,Ann. Inst. Fourier (Grenoble) 31:3 (1981), 169–223.

    MathSciNet  Google Scholar 

  14. Helffer, B. andRobert, D., Calcul fonctionnel par la transformation de Melline et opérateurs admissibles,J. Funct. Anal. 53 (1983), 246–268.

    Article  MathSciNet  Google Scholar 

  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).

  16. Hislop, P. andNakamura, S., Semiclassical resolvent estimates,Ann. Inst. H. Poincaré Phys. Théor. 51 (1989), 187–198.

    MathSciNet  Google Scholar 

  17. Jecko, T., Sections efficaces totales d'une molécule diatomique dans l'approximation de Born-Oppenheimer, Thesis, Nantes, 1996.

  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.

    MathSciNet  Google Scholar 

  19. Robert, D.,Autour de l'approximation semi-classique, Progr. Math.68, Birkhäuser, Boston, Mass., 1987.

    Google Scholar 

  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.

    MATH  Google Scholar 

  21. Robert, D., On the scattering theory for long range perturbations of Laplace operators,J. Anal. Math. 59 (1992), 189–203.

    MATH  MathSciNet  Google Scholar 

  22. Robert, D., Relative time-delay for perturbations of elliptic operators and semiclassical asymptotics,J. Funct. Anal. 126 (1994), 36–82.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MathSciNet  Google Scholar 

  24. Thaller, B.,The Dirac Equation, Texts and Monographs in Phys., Springer-Verlag, Berlin-Heidelberg-New York, 1992.

    Google Scholar 

  25. Yajima, K., The quasi-classical approximation to Dirac equation, I,J. Fac. Sci. Univ. Tokyo Sect. I A Math. 29 (1982), 161–194.

    MATH  MathSciNet  Google Scholar 

  26. Yamada, O., On the principle of limiting absorption for the Dirac operators,Publ. Res. Inst. Math. Sci. 8 (1972/73), 557–577.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de mathématiques, Université Bordeaux I, 351, cours de la Libération, F-33405, Talence, France

    Vincent Bruneau

  2. Département de mathématiques, Université de Nantes, 2, rue de la Houssinière, BP 92208, F-44322, Nantes Cedex 03, France

    Didier Robert

Authors
  1. Vincent Bruneau
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Didier Robert
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bruneau, V., Robert, D. Asymptotics of the scattering phase for the Dirac operator: High energy, semi-classical and non-relativistic limits. Ark. Mat. 37, 1–32 (1999). https://doi.org/10.1007/BF02384826

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02384826

Keywords

Search

Navigation