Pseudoharmonic Oscillators and Inadequacy of Semiclassical Quantization.

A fundamental property of harmonic oscillators is that their classical motions have energy-independent periods. This property is not confined to harmonic oscillators, but includes a broad family of anharmonic cases as well. The Einstein-Brillouin- Keller prescription for semiclassical quantization leads to the familiar harmonic oscillator spectrum for all of these "pseudoharmonic oscillators." We have explicitly constructed a class of pseudoharmonic oscillators that are free of singularities along the real axis. A limiting case for this class is the split harmonic oscillator (unequal left and right side harmonic forces), for which exact Schrodinger equation eigenvalues have been obtained. The latter deviate significantly from predictions of the semiclassical approximations in regards to (a) zero-point energy of the ground state, (b) separation between successive eigenvalues, and (c) oscillatory behavior of the energies of highly excited states as the extent of anharmonicity varies.