  In a previous paper we solved a countably infinite family of one-dimensional
Schr\"odinger equations by showing that they were supersymmetric partner
potentials of the standard quantum harmonic oscillator. In this work we extend
these results to find the complete set of real partner potentials of the
harmonic oscillator, showing that these depend upon two continuous parameters.
Their spectra are identical to that of the harmonic oscillator, except that the
ground state energy becomes a tunable parameter. We finally use these
potentials to analyse the physical problem of Bose-Einstein condensation in an
atomic gas trapped in a dimple potential.
