Free Propagator Expansion in the Evaluation of the Lamb Shift. I

Abstract

This is the first of two papers concerned with ``order properties,'' with respect to the parameter αZ, of an expansion method for the evaluation of the bound electron self-energy ΔE and the application of these properties to the calculation of the new Lamb shift orders of α(αZ)6 ln2(αZ) and α(αZ)6 ln(αZ). The expansion method is the free-propagator expansion (FPE); that is, the formal algebraic expansion of the bound electron propagator or Green's function in ``powers'' of the external (Coulomb) potential. The principal result of the general mathematical analysis is a theorem which asserts that the FPE is an order expansion for (only) those terms of ΔE that are nonanalytic in the parameter w ≡ (αZ)2 and is thus particularly suitable for the calculation of this class of terms. A practical result of the theorem is that the new logarithmic orders arise from only the first four terms of the FPE. The nonanalytic part of a fixed term In of the FPE can be attacked directly through a consideration of Im+In, where Im+In denotes the imaginary part of In, regarded as a function of the complex variable w, on the upper side of a branch cut along the negative w axis. As an auxiliary result, boundedness properties in momentum space are derived for certain iterated operators related to the FPE of the bound nonrelativistic electron Green's function.