The infrared problem for the dressed non-relativistic electron in a magnetic field

We consider a non-relativistic electron interacting with a classical magnetic field pointing along the x3-axis and with a quantized electromagnetic field. The system is translation invariant in the x3-direction and the corresponding Hamiltonian has a decomposition H ≅∫R+H(P3)dP3. For a fixed momentum P3 sufficiently small, we prove that H(P3) has a ground state in the Fock representation if and only if E'(P3)=0, where P3 →E'(P3) is the derivative of the map P3→E(P3)=infσ(H(P3)). If E'(P3)≠0, we obtain the existence of a ground state in a non-Fock representation. This result holds for sufficiently small values of the coupling constant. (authors)