Landau damping in the transverse plane

In these proceedings we sketch a general theory for Landau damping in the transverse plane, in the particular case of beam-coupling impedance with linear (octupolar) detuning as the source of tune spread. Using a Hamiltonian formalism and perturbation theory, we will go beyond the traditional stability diagram approach to obtain the general non-linear determinant equation that leads to modes determination. Limiting cases are studied, and preliminary results of the full formalism are obtained in an attempt to generalize the concept of stability diagram.