The projected single-particle Dirac operator for Coulombic potentials

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 $\Lambda_+ A\,\Lambda_+\,+\Lambda_-A\,\Lambda_-\;$ to any given order in the potential strength, where $\Lambda_+$ and $\Lambda_-$ project onto the positive and negative spectral subspaces of the free Dirac operator. To first order, $\Lambda_+ A \,\Lambda_+$ 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 $p$-form bounded. In the case of a modified Coulomb potential, $V=-\gamma x^{-1+\epsilon},\;\;\epsilon >0,\;$ one can even prove subordinacy of the $n$-th order term in $\gamma$ with respect to the $n-1$st order term for all $n>1$, as well as their $p$-form boundedness with form bound less than one.