Theoretical models for longitudinal coupled-bunch instabilities driven by harmonic cavities in electron storage rings

We present a theoretical framework for analyzing longitudinal coupled-bunch instabilities in double-rf systems with even filling patterns, accounting for potential-well distortion and multiple azimuthal modes. The linearized Vlasov equation is solved in the frequency-domain for an arbitrary rf potential to derive the Lebedev equation. We unified different formulations, obtaining results from recent publications as particular cases. Applications to Robinson dipole-quadrupole mode coupling and the periodic transient beam loading (PTBL)/mode-1 instability are presented. Notably, for the first time, theoretical predictions of the mode-1 thresholds show excellent agreement with experimental data. The analysis reveals that the PTBL instability is a zero-frequency effect dependent on azimuthal mode interactions and resistant to Landau damping, providing new insights into its mechanism. The methods are implemented in the open-source package pycolleff, offering a useful semi-analytical tool for studying instabilities in electron storage rings with harmonic cavities.