Jump in adiabatic invariant at a transition between modes of motion for systems with impacts

The one-dimensional motion of a particle between slowly moving reflecting walls in a slowly varying potential is considered. It is shown that standard adiabatic perturbation theory describes dynamics in the mode with impacts with the same accuracy as in the mode without impacts. An asymptotic formula for the jump of adiabatic invariant due to the change in mode of the motion is obtained.