The limit of classical orbits approaching for the Coulomb potential (left), the SHO potential (center), and the ISW potential (right). The parameter captures the number of times a path between the maximum radius and the center is traversed in the limit ; for the Coulomb potential this is , for the other two it is .
The numerical energy spectrum of the logarithmic potential plotted with the old quantization spectrum from Eq. (23) . The residual differences between the two are shown in the inset.
The first 50 critical values , plotted with the prediction using old quantization from Eq. (29) . The residual differences between the two are shown in the inset.
The first 20 numerically calculated energy eigenvalues for spherically-symmetric states in the logarithmic potential, along with the first 20 old-quantization energy values. Numerical errors are on the order of , which is generally much smaller than the discrepancy between the numerical eigenvalue and the old-quantization eigenvalue.
The first 20 numerically calculated critical values , together with the old-quantization values. Numerical errors range from to and are in general much smaller than the discrepancy between old-quantization and numerical values.
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