proceeds via perturbation theory. First order perturbation theory will predict an energy of the order of $\alpha \Lambda ^2$, and a higher order power counting argument confirms the asymptotically large $\Lambda$ dependence of that calculation. Our theorems below show that the predictions of perturbation theory for the self–energy problem are wrong, if one is interested in the large $\Lambda$ asymptotics of the energy. If perturbation theory works at all, then it works only for a range of $\alpha$ that vanishes as $\Lambda$ increases. In fact we deduce from the upper bound in Theorem \[theo:nonrel\] that the size of this range shrinks at least as $\Lambda^{-2/5}$.

All the theorems below are asymptotic statements for large $\Lambda$ and for fixed $\alpha$. For actual bounds we refer the reader to \[LL\]. The first result concerns the self energy of a nonrelativistic electron interacting with the radiation field. The Hamiltonian is given by

$$H= \frac{1}{2} (p+ \sqrt{\alpha} A(x))^2 +H_{f} \ ,
\label{bacon}$$

where $p =-i\nabla$ and acts on $L^2(\mathbb{R}^3)\otimes {\mathcal F}$, where ${\mathcal F}$ denotes the photon Fock space.

The ground state energy, $E_0$, of the operator (\[bacon\]) satisfies the bounds $$C_1 \alpha^{1/2} \Lambda^{3/2} \ < \ E_0 \ <  \ C_2 
\alpha^{2/7} \Lambda^{12/7}
