  The general question of how a beam becomes unstable has been one of the
fundamental research topics among beam and accelerator physicists for several
decades. In this study, we revisited the general problem of linear beam
stability in periodic focusing systems by applying the concepts of Krein
signature and band structure. We numerically calculated the eigenvalues and
other associated characteristics of one-period maps, and discussed the
stability properties of single-particle motions with skew quadrupoles and
envelope perturbations in high-intensity beams on an equal footing. In
particular, an application of the Krein theory to envelope instability analysis
was newly attempted in this study. The appearance of instabilities is
interpreted as the result of the collision between eigenmodes of opposite Krein
signatures and the formation of a band gap.
