Method of Linear Invariants for description of beam dynamics of FEL   undulator
  We propose a new model for description of electrons beam dynamics in Free Electron Laser (FEL) undulator, based on the method of linear time-dependent invariants of quantum-mechanical charge particle. The magnetic field has periodic structure along the undulator. For this problem, described by time-dependent quadratic Hamiltonian, we obtain exact solution. The time-evolutions of the tree quantum fluctuations: covariance cov(q,p), var(q) and var(p) for the charge particle in this case are also determined. This research will help to optimize the FEL undulator: for example, using a 2.5 GeV linear electron accelerator it will be possible to emit radiation at 1.5 nm and shorter length.   This method could be applicable also to any device with periodic structure of applied field (e.g. Tokamak, cyclic accelerators) for the case of charge non-relativistic quantum particles. 
A. Angelow, D. Trifonov, V. Angelov, H. Hristov