Sufficient criteria for stability and narrowness of the band-shaped ion beams in 3d electric and magnetic fields with plane of symmetry

The problem of stability of the charged particle beams is considered when they are shifted outside the plane of symmetry z = 0 of the electric and/or magnetic field. It is shown that if the electrostatic potential U(x, y) defined along the plane of symmetry z = 0 satisfies the relation Uxx + Uyy