One-loop self-energy correction to the1sand2shyperfine splitting in H-like systems

The one-loop self-energy correction to the hyperfine splitting of the $1s$ and $2s$ levels in H-like low-$Z$ atoms is evaluated to all orders in $Z\alpha .$ The results are compared to perturbative calculations. The residual higher-order contribution is evaluated. Implications to the specific difference of the hyperfine-structure intervals $8\Delta {\nu }_2-\Delta {\nu }_1$ in ${\mathrm{He}}^{+}$ are investigated.