Expansion of Bound-State Energies in Powers ofm/M

We describe a new approach to computing energy levels of a nonrelativistic bound state of two constituents with masses $M$ and $m$, by a systematic expansion in powers of $m/M$. After discussing the method, we demonstrate its potential with an example of the radiative recoil corrections to the Lamb shift and hyperfine splitting relevant for the hydrogen, muonic hydrogen, and muonium. A discrepancy between two previous calculations of $O\left(\alpha \right(Z\alpha {)}^5{m}^2/M)$ radiative recoil corrections to the Lamb shift is resolved and several new terms of $O\left(\alpha \right(Z\alpha {)}^5{m}^4{/M}^3)$ and higher are obtained.