Equilibrium Distribution and Tune Shift of Beams in a Linear Collider
  The evolution of two colliding beams in a linear accelerator can be described by two coupled Vlasov equations. In PEP-NOTE-325 by S. Kheifets and A. W. Chao the case without external focusing was considered. In this paper we derive the equilibrium distribution and the tune shift in the presence of external focusing. Motion is considered only in the vertical direction and the beams are presumed to be one-dimensional. 
Bjoern S. Schmekel and Joseph T. Rogers