The self-energy of one electron in non relativistic QED

We investigate the self-energy of one electron coupled to a quantized radiation field by extending the ideas developed previously by Hainzl. We fix an arbitrary cutoff parameter $\Lambda$ and recover the $\alpha^2$-term of the self-energy, where $\alpha$ is the coupling parameter representing the fine structure constant. Thereby we develop a method which allows to expand the self-energy up to any power of $\alpha$. This implies that perturbation theory in $\alpha$ is correct if $\Lambda$ is fix. As an immediate consequence we obtain enhanced binding for electrons.ou