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Potentials with Two Shifted Sets of Equally Spaced Eigenvalues and Their Calogero Spectrum

Motivated by the concept of shape invariance in supersymmetric quantum mechanics, we obtain potentials whose spectrum consists of two shifted sets of equally spaced energy levels. These potentials are similar to the Calogero-Sutherland model except the singular term $αx^{-2}$ always falls in the transition region $-1/4 < α< 3/4$ and there is a delta-function singularity at x=0.