The Projected Single-Particle Dirac Operator for Coulombic Potentials

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+ + Λ−AΛ− to any given order in the potential strength, where Λ+ and Λ− project onto the positive and negative spectral subspaces of the free Dirac operator. To first order, Λ+AΛ+ 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 = −γx−1+ , > 0, one can even prove subordinacy of the n-th order term in γ with respect to the n − 1st order term for all n > 1, as well as their p-form boundedness with form bound less than one. 2000 Mathematics Subject Classification: 81Q10, 81Q15, 81V45