Approximate numerical calculation of the self-energy of a bound electron Original Research Article

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 image, image and image states of the hydrogen-like ions. It is for the first time the results for image states for image from 5 to 55 are presented. For image states and for nuclear charges up to image 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 image 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 image state of the hydrogen atom is 4 parts in image. Reliable predictions for the self-energy are obtained up to image 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.