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The Bertrand's theorem is extended, i.e. closed orbits still may exist forother central potentials than the power law Coulomb potential and isotropicharmonic oscillator. It is shown that for the combined potential$V(r)=W(r)+b/r^2$ ($W(r)=ar^{\nu}$), when (and only when) $W(r)$ is the Coulombpotential or isotropic harmonic oscillator, closed orbits still exist forsuitable angular momentum. The correspondence between the closeness ofclassical orbits and the existence of raising and lowering operators derivedfrom the factorization of the radial Schr\"odinger equation is investigated.Comment: 4 pages, 1 figug

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