Perturbed Orbital Contribution to the Two-Loop Lamb Shift in Hydrogen

A part of the two-loop Lamb shift called the perturbed orbital term is evaluated with exact Dirac-Coulomb propagators. It is shown to be the most nonperturbative function of $Z\alpha$ yet encountered in QED, so much so that even at $Z=1\mathrm{}$ the leading term in its $Z\alpha$ expansion is of the opposite sign from the complete answer. The higher order contributions are $-71\mathrm{}\left(1\right)\mathrm{kHz}$ to the ground state Lamb shift in hydrogen and $-8.9\mathrm{}\left(1\right)$ and $-346\mathrm{}\left(1\right)\mathrm{kHz}$ to the $n=2\mathrm{}$ Lamb shift in hydrogen and ${\mathrm{He}}^{+}$, respectively.