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 $α$, the leading order term is represented by $2π^{-1} α(Λ- \ln[1 + Λ])$. Next we put the electron in the field of an arbitrary external potential $V$, such that the corresponding Schrödinger 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 $α$. Moreover, we provide concrete numbers for $α$, the ultraviolet cut-off $Λ$, and the radiative correction for which our procedure works.

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