Extension of Bertrand’s theorem and factorization of the radial Schrödinger equation

The orbit of a classical particle in a central field, due to the angular momentum conservation, must lie in a plane perpendicular to the angular momentum. However, the orbit is, in general, not closed. In classical mechanics there is a famous Bertrand’s theorem [1,2], which says that the only central forces that result in closed orbits for all bound particles are the inverse square law and Hooke’s law. For the two central potentials, apart from the energy and angular momentum, there exists an additional conserved quantity, which implies a higher dynamical symmetry than the geometrical symmetry (space isotropy). Over fifty years ago, Schrödinger introduced the factorization method [3] to treat the eigenvalue problem of a one-dimensional harmonic oscillator and the quantized energy eigenvalues of a harmonic oscillator are connected by energy raising and lowering operators. The factorization method was generalized in the supersymmetry quantum mechanics (SSQM) [4] with the help of the concept of supersymmetry and shape invariance. SSQM mainly focuses on the factorization of one-dimensional Schrödinger equation (including the radial Schrödinger equation of a particle in a central field) [5] and the relation of eigenstates between two quantum systems (supersymmetric partner). It was shown [6,7] that only for the Coulomb potential and isotropic harmonic oscillator the radial Schrödinger equation can be factorized and both the angular momentum and energy raising and lowering operators can be constructed. This reminds us that there may exist a certain connection between the factorization of radial Schrödinger equation and the closeness of classical orbits. Careful examination shows that in the derivation of Bertrand’s theorem and in [6], a power law central potential (V (r) = ar) was assumed. For such a power law central potential, the Bertrand’s theorem does hold. However, more careful analysis shows that if the restriction of power law form central potential is relaxed, the Bertrand’s theorem need reexamination. In sect. II, it is shown that for the combined type of central potential V (r) = W (r) + b/r (W (r) = ar), when W (r) is the Coulomb potential or isotropic harmonic potential, there still exist closed orbits for suitable angular momenta. The factorization of the corresponding radial Schrödinger equation and the connection between the closeness of classical orbits and the existence of raising and lowering operators are investigated in sect. III. A brief summary is given in sect. IV.