Single bunch monopole instability

We study single bunch stability with respect to monopole longitudinal oscillations in electron storage rings. Our analysis is different from the standard approach based on the linearized Vlasov equation. Rather, we reduce the full nonlinear Fokker-Planck equation to a Schrodinger-like equation which is subsequently analyzed by perturbation theory. We show that the Haissinski solution (1973) may become unstable with respect to monopole oscillations and derive a stability criterion in terms of the ring impedance.