  We consider the infrared problem in a model of a freely propagating,
nonrelativistic charged particle of mass 1 in interaction with the quantized
electromagnetic field. The hamiltonian of the system is regularized by an
infrared cutoff $\ssig\ll 1$ and an ultraviolet cutoff $\Lambda\sim 1$ in the
interaction term, in units of the mass of the charged particle. Due to
translation invariance, it suffices to study $\Hps$, the restriction of the
hamiltonian to the fibre Hilbert space of the conserved momentum operator
associated to total momentum $p\in\R^3$. Under the condition that the coupling
constant $g$ and the conserved momentum $p$ are sufficiently small, the
following statements hold: (1) For every $\ssig>0$, $E_0:=inf spec \Hps$ is an
eigenvalue with corresponding eigenvector $\Omega[p,\ssig]\in\Hp$. (2) For all
$\ssig\geq0$, the first and second derivatives of $(\Egrd-\frac{|p|^2}{2})$ are
$O(g^\delta)$ for some $\delta>0$. (3) $\Omega[p,\ssig]$ is not an element of
the Fock space $\Hp$ in the limit $\ssig\to0$, if $|p|>0$. Our proofs are based
on the operator-theoretic renormalization group of V. Bach, J. Fr\"ohlich, and
I.M. Sigal. The key difficulty in the analysis of this system is connected to
the strictly marginal nature of the leading interaction term, and a main issue
in our exposition is to develop analytic tools to control its renormalization
flow.
