Resonances of electron beam focused by helical quadrupole magnetic field

The conditions have been found for the instability of waves in an electron beam to be focused by a homogeneous longitudinal magnetic field and spiral-like quadrupole one both. The system is confined by a waveguide. The properties of waves of various types are analyzed within framework of the perturbation theory. The possibility of the microwave generation due to instabilities is discussed. The increments of instabilities are estimated.