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 2alphapi(-1) (Lambda - ln[1 + Lambda]). Next we put the electron in the field of an arbitrary external potential V, such that the corresponding Schrodinger 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 a, the ultraviolet cut-off Lambda, and the radiative correction for which our procedure works.