Coupled Landau damping with elliptical spectra

In this report, the criterion for Landau damping in the presence of linear coupling between the transverse planes, previously obtained for Lorentz betatron frequency distributions , is extended to "elliptical" spectra . Different limiting cases are discussed and a general stability criterion is derived. The results reveal two important features of coupled Landau damping. The first one is that the effect of the imaginary part of the coupling impedance in the unstable plane, the most critical parameter for Landau damping at low and medium energy in the absence of coupling, can be "cancelled" by coupling. The second feature is that the frequency spreads are "shared" between the two planes and a large tune dispersion becomes effective also in the plane which, without coupling, has little spread. The conjunction of these two features explains why a machine like the PS can be stabilised by tuning close to a coupling resonance and determining values for the optimum coupling.