Space charge effects play significant role in modern-day accelerators. These effects frequently constrain attainable beam parameters in an accelerator - or - in an accelerator chain. They also could limit the luminosity of hadron colliders operating either at low energies or with a sub-TeV high brightness hadron beams. A number of schemes for compensating space charge effects in a coasting (e.g. continuous) hadron beam were proposed and some of them had been tested. But all of these methods do not address the issue of tune spread compensation of a bunched hadron beam, e.g. the tune shift dependence on the longitudinal position inside the bunch. In this paper we propose and evaluate a novel idea of using a co-propagating electron bunch with miss-matched longitudinal velocity to compensate the space charge induced tune-shift and tune spread. We present a number of practical examples of such system.

40 pages, 12 figures