Stability of periodically focused intense particle beams

A stability analysis of periodically focused intense particle beams based on the beam envelope equation is performed. We show that (i) the scenario, as the focusing field increases, is not the existence of a single threshold above which stable matched (equilibrium) solutions are absent, as generally believed, but the existence of successive regions of stability interrupted by gaps of instability; (ii) the beam can be focused to tighter radii using new stable matched solutions found for focusing field strengths greater than the previous threshold. Self-consistent simulations validate the findings.