One non-relativistic particle coupled to a photon field

We investigate the ground state energy of an electron coupled to a photon field. First, we regard the self-energy of a free electron, which we describe by the Pauli-Fierz Hamiltonian. We show that, in the case of small values of the coupling constant $\alpha$, the leading order term is represented by $2\pi^{-1} \alpha (\Lambda - \ln[1 + \Lambda])$. Next we put the electron in the field of an arbitrary external potential $V$, such that the corresponding Schr\"odinger operator $p^2 + V$ has at least one eigenvalue, and show that by coupling to the radiation field the binding energy increases, at least for small enough values of the coupling constant $\alpha$. Moreover, we provide concrete numbers for $\alpha$, the ultraviolet cut-off $\Lambda$, and the radiative correction for which our procedure works.