  We developed a theory of electron scattering by a short-range repulsive
potential in a cavity. In the regime of ultrastrong electron coupling to the
cavity electromagnetic field, the vacuum fluctuations of the field result in
the dynamical stabilization of a quasistationary polariton state confined in
the core of the repulsive potential. When the energy of a free electron
coincides with the energy of the confined state, the extremely efficient
resonant nonelastic scattering of the electron accompanied by emission of a
cavity photon appears. This effect is discussed as a basis for possible
free-electron sources of nonclassical light.
