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


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).


Electromagnetic Field Euclidean Space Asymptotic Expansion Accurate Result Technical Problem 
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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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