The dressed nonrelativistic electron in a magnetic field

We consider a nonrelativistic electron interacting with a classical magnetic field pointing along the x3-axis and with a quantized electromagnetic field. When the interaction between the electron and photons is turned off, the electronic system is assumed to have a ground state of finite multiplicity. Because of the translation invariance along the x3-axis, we consider the reduced Hamiltonian associated with the total momentum along the x3-axis and, after introducing an ultraviolet cutoff and an infrared regularization, we prove that the reduced Hamiltonian has a ground state if the coupling constant and the total momentum along the x3-axis are sufficiently small. We determine the absolutely continuous spectrum of the reduced Hamiltonian and, when the ground state is simple, we prove that the renormalized mass of the dressed electron is greater than or equal to its bare one. We then deduce that the anomalous magnetic moment of the dressed electron is nonnegative.