Extension Of Bertrand's Theorem And Factorization Of The Radial Schrödinger Equation

Details Extension Of Bertrand's Theorem And Factorization Of The Radial Schrödinger Equation Extension Of Bertrand's Theorem And Factorization Of The Radial Schrödinger Equation

1998-12-27

arxiv.org/abs/quant-ph/9812077v1

J.Math.Phys. 39 (1998) 5253-5259

Physics

Quantum Physics

4 pages, 1 figuge

Scientific paper

10.1063/1.532568

The Bertrand's theorem is extended, i.e. closed orbits still may exist for other central potentials than the power law Coulomb potential and isotropic harmonic 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 Coulomb potential or isotropic harmonic oscillator, closed orbits still exist for suitable angular momentum. The correspondence between the closeness of classical orbits and the existence of raising and lowering operators derived from the factorization of the radial Schr\"odinger equation is investigated.

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