  Modern and future particle accelerators employ increasingly higher intensity
and brighter beams of charged particles and become operationally limited by
coherent beam instabilities. Usual methods to control the instabilities, such
as octupole magnets, beam feedback dampers and use of chromatic effects, become
less effective and insufficient. We show that, in contrast, Lorentz forces of a
low-energy, a magnetically stabilized electron beam, or "electron lens", easily
introduces transverse nonlinear focusing sufficient for Landau damping of
transverse beam instabilities in accelerators. It is also important that,
unlike other nonlinear elements, the electron lens provides the frequency
spread mainly at the beam core, thus allowing much higher frequency spread
without lifetime degradation. For the parameters of the Future Circular
Collider, a single conventional electron lens a few meters long would provide
stabilization superior to tens of thousands of superconducting octupole
magnets.
