The two-loop self-energy correction to the ground-state energy levels of hydrogen-like ions with nuclear charges Z ≥10 is calculated without the Z α expansion, where α is the fine-structure constant. The data obtained are compared with the results of analytical calculations within the Z α expansion; significant disagreement with the analytical results of order α 2 ( Z α) 6 has been found. Extrapolation is used to obtain the most accurate value for the two-loop self-energy correction for the 1 s state in hydrogen.
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Yerokhin et al. (2005) studied this question.
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