Simple proof of electron beam stability in the deep-potential-well equilibrium

It is well known that a strongly magnetized electron beam propagating in an evacuated conducting pipe can exist in either of two equilibria, one with a shallow potential well and the other with a deep potential well. A Lorentz transformation connecting the two classes of equilibria is derived. Thus, since shallow-potential-well equilibria are known to be stable, the deep-potential-well equilibria must also be. The stability of space charge waves in the two equilibria should follow the same pattern.<
>