Quantization of the multifold Kepler system

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For over a century, the Kepler problem and the harmonic oscillator have been known as the only central force dynamical systems, all of whose bounded motions are periodic. Two of the authors (T. I. and N. K.) have found an infinite number of dynamical systems possessing such a periodicity property, which have been called multi-fold Kepler systems or ν-fold Kepler systems, with ν a positive rational number. If ν is allowed to take the real positive numbers, say ν=α, then for the α-fold Kepler system, all the bounded motions become periodic or not, according to whether the parameter α is a rational number or not. A purpose of this paper is to quantize the α-fold Kepler system and thereby to figure out a quantum analog of the closed orbit property of the α-fold Kepler system. It will turn out that the quantized α-fold Kepler system admits accidental degeneracy in energy levels or not, according to whether α is a rational number or not.