Equally-spaced energy spectra: the harmonic oscillator with a centrifugal barrier or with a centripetal well

We examine the spectrum and eigenfunctions of the quantum harmonic oscillator with a centrifugal barrier and with a centripetal well. Three intervals in the centrifugal-centripetal strength parameter range are relevant: in one the spectrum is unique, lower-bound, and equally-spaced, in the second it is lower-bound but not unique and - except for two cases - not equally spaced, while in the third it has neither of the three properties. An associated two-component Hamiltonian has a spectrum exhibiting equal spacing over the full parameter range. (author)