Closure of orbits and dynamical symmetry of screened Coulomb potential and isotropic harmonic oscillator

It is shown that for any central potential V(r) there exist a series of conserved aphelion and perihelion vectors R-tilde=pxL-g(r)r, g(r)=rV'(r). However, if and only if V(r) is a pure or screened Coulomb potential, R-tilde and L constitute an SO4 algebra in the subspace spanned by the degenerate states with a given energy eigenvalue E<0 at the aphelia and perihelia (r=0). For a pure Coulomb potential, R-tilde is reduced to the Pauli-Runge-Lenz (PRL) vector R and for a screened Coulomb potential R-tilde is reduced to the extended PRL vector R'. While dR/dt=0 always holds, dR'/dt=0 holds only at the aphelia and perihelia. Moreover, the space spanning the SO4 algebra for a screened Coulomb potential is smaller than that for a pure Coulomb potential. The relation of closed orbits for a screened Coulomb potential with that for a pure Coulomb potential is clarified. The ratio of the radial frequency ωr and angular frequency ωφ, ωr/ωφ=κ=1 for a pure Coulomb potential irrespective of the angular momentum L and energy E(<0). For a screened Coulomb potential κ is determined by the angular momentum L, and when κ is any rational number (κ<1), the orbit is closed. The situation for a pure or screened isotropic harmonic oscillator is similar