Proof that stable monotonic equilibrium distributions in a continuous-focusing channel are necessarily axisymmetric

The transverse Vlasov equilibrium distribution function of an unbunched ion beam propagating in a continuous-focusing channel is specified by a function f_⊥(H_⊥), where H_⊥ is the single-particle Hamiltonian. In standard treatments of continuous-focusing equilibria in Vlasov-Poisson electrostatic models, it is assumed that a stable beam equilibrium specified by monotonic f_⊥(H_⊥) with ∂f_⊥(H_⊥)/∂H_⊥≤0 is axisymmetric (no variation in azimuthal angle, i.e., with ∂/∂θ=0). In this paper a simple, but rigorous, proof is presented that only axisymmetric equilibrium solutions are possible in Vlasov-Poisson models for any physical choice of f_⊥(H_⊥) with ∂f_⊥(H_⊥)/∂H_⊥≤0 if the confining boundary of the system (the beam pipe) is axisymmetric or if the geometry is radially unbounded.