A new mechanism of particle acceleration and rotation numbers

A mathematical model based on real physical processes in which there is unbounded growth of the energy of particles in an arbitrarily small region of space is proposed. The model is a generalization of the relativistic analog of the Fermi–Ulam model [1]. The generalization is that after collision with the lower wall the particle moves in the same way as after collision with a third horizontal infinitely heavy wall that moves in the vertical direction in accordance with a periodic law. It is shown that for conditions of general form the energy of the particle increases to infinity for the majority of the initial data. An asymptotic lower bound for this growth is established, and examples are found for which unlimited growth of the particle energy is possible for arbitrarily small separation between the walls.