Two-loop self-energy contribution to the Lamb shift in H-like ions

The two-loop self-energy correction is evaluated to all orders in $Z\ensuremath{\alpha}$ for the ground-state Lamb shift of H-like ions with $Z\ensuremath{\geqslant}10$, where $Z$ is the nuclear charge number and $\ensuremath{\alpha}$ is the fine-structure constant. The results obtained are compared with the analytical values for the $Z\ensuremath{\alpha}$-expansion coefficients. An extrapolation of the all-order numerical results to $Z=1$ is presented and implications of our calculation for the hydrogen Lamb shift are discussed.