On the Infrared Problem for the Dressed Non-Relativistic Electron in a Magnetic Field

Abstract. 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 we consider the reduced Hamiltonian H(P3) associated with the total momentum P3 along the x3-axis. 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 7→ E ′(P3) is the derivative of the map P3 7→ E(P3) = inf σ(H(P3)). If E′(P3) 6= 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. MSC: 81V10; 81Q10; 81Q15