The projected single-particle Dirac operator for Coulombic potentials

  • D.H. Jakubassa-Amundsen

    Mathematics Institute University of Munich Theresienstr. 39 80333 Munich Germany
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Abstract

A sequence of unitary transformations is applied to the one-electron Dirac operator in an external Coulomb potential such that the resulting operator is of the form Λ+AΛ++ΛAΛ  \Lambda_+ A\,\Lambda_+\,+\Lambda_-A\,\Lambda_-\; to any given order in the potential strength, where Λ+\Lambda_+ and Λ\Lambda_- project onto the positive and negative spectral subspaces of the free Dirac operator. To first order, Λ+AΛ+\Lambda_+ A \,\Lambda_+ coincides with the Brown-Ravenhall operator. Moreover, there exists a simple relation to the Dirac operator transformed with the help of the Foldy-Wouthuysen technique. By defining the transformation operators as integral operators in Fourier space it is shown that they are well-defined and that the resulting transformed operator is pp-form bounded. In the case of a modified Coulomb potential, V=γx1+ϵ,    ϵ>0,  V=-\gamma x^{-1+\epsilon},\;\;\epsilon >0,\; one can even prove subordinacy of the nn-th order term in γ\gamma with respect to the n1n-1st order term for all n>1n>1, as well as their pp-form boundedness with form bound less than one.

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D.H. Jakubassa-Amundsen, The projected single-particle Dirac operator for Coulombic potentials. Doc. Math. 10 (2005), pp. 331–356

DOI 10.4171/DM/191