Computation of longitudinal bunched beam instability thresholds

An integral equation derived from the linearized Vlasov equation has been used to find the instability thresholds in the case of space-charge impedance alone for various distribution functions. It has been found that the thresholds for the instability which are caused by the coupling between m=±1 azimuthal modes may be obtained analytically for many practically used distributions. Moreover, the criterion determining these thresholds appears to be the same as that for thresholds beyond which no stationary distribution can be found