From wikipedia And my lecture notes have a derivation for the final photon energy in inverse compton scattering where the photon energy before and after the collision are the same in the rest frame of the electron. The change in photon energy merely comes from transforming in and out of the electron's rest frame. But surely the photon energy is not the same before and after the collision in the frame where the electron is initially at rest. There is a shift in the wavelength and the electron gains kinetic energy in that frame. So I don't see how the above is valid.
Hmm, I see your problem. I think the problem lies in considering the scattering of just one photon at a time. Thompson scattering is a classical description of scattering and I think in that derivation the incident radiation field gets scattered off in all different directions by the election without any net transfer of energy to or from the electron (in it's rest frame). Of course you can't do this with just one photon so I think it is a result of averaging over many scattering events, although you don't think about it that way in the classical description.
