  We consider a nonrelativistic electron interacting with a classical magnetic
field pointing along the $x_{3}$-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 $x_{3}$-axis, we consider the
reduced Hamiltonian associated with the total momentum along the $x_{3}$-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 $x_{3}$-axis are sufficiently small. Finally
we determine the absolutely continuous spectrum of the reduced Hamiltonian.
