Approximate numerical calculation of the self-energy of a bound electron

The evaluation method of the self-energy of the bound electron in the one-loop approximation is described. It is based on the expansion of physical momentum of the electron around the momentum of the electron at rest. The method is applied to 1S1S, 2P3/22P3/2 and 3D5/23D5/2 states of the hydrogen-like ions. It is for the first time the results for DD states for ZZ from 5 to 55 are presented. For PP states and for nuclear charges up to ZZ about 40 the accuracy of the results is comparable to that of the most accurate calculations made so far. If the exact value of the coefficient A50A50 in the expansion of the self-energy is used as an additional input, the relative difference between the present method and the most accurate value given so far in the literature for 1S1S state of the hydrogen atom is 4 parts in 106106. Reliable predictions for the self-energy are obtained up to ZZ about 90 for all states. The calculation is analytic up to the one-dimensional integration over continuous part of the hydrogen spectrum. The separation of the low and high energy regions and their matching is avoided.

► The results for 3D5/23D5/2 states for Z∈(5,55)Z∈(5,55) are given here for the first time.
► The results for 2P3/22P3/2 states up to ZZ about 40 are very accurate.
► Reliable predictions are obtained up to ZZ about 90 for all states.
► Up to 1D integration the calculation is analytic.
► The separation into the low and high energy regions and their matching are avoided.