Progress on the Intensity Frontier of high energy physics critically depends on record high intensity charged particles accelerators. Beams in such machines become operationally limited by coherent beam instabilities, particularly enhanced in the regime of strong space charge (SC). Usual methods to control the instabilities, such as octupole magnets, beam feedback dampers and employment of chromatic effects, become less effective and insufficient. In [1] it was proposed to employ electron lenses for introduction of sufficient spread in particle oscillation frequencies needed for beam stabilization and in [2] it was shown that electron lenses are uniquely effective for Landau damping of transverse beam instabilities in high energy particle accelerators and their employment does not compromise incoherent (single particle) stability, dynamic aperture and the beam lifetime. Here we consider an important issue of effectiveness of the Landau damping with electron lenses in space-charge dominated beams and demonstrate that the desired stability can be assured with proper choice of the electron beam parameters and current distributions.
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