Holomorphy of the scattering matrix with respect toc −2 for Dirac operators and an explicit treatment of relativistic corrections

We prove holomorphy of the scattering matrix at fixed energy with respect toc^–2 for abstract Dirac operators. Relativistic corrections of orderc^–2 to the nonrelativistic limit scattering matrix (associated with an abstract Pauli Hamiltonian) are explicitly determined. As applications of our abstract approach we discuss concrete realizations of the Dirac operator in one and three dimensions and explicitly compute relativistic corrections of orderc^–2 of the reflection and transmission coefficients in one dimension and of the scattering matrix in three dimensions. Moreover, we give a comparison between our approach and the firstorder relativistic corrections according to Foldy-Wouthuysen scattering theory and show complete agreement of the two methods.Supported by Fonds zur Förderung der wissenschaftlichen Forschung in Österreich by an E. Schrödinger Fellowship and by Project No. P7425