In a previous paper (Fellows and Smith 2009 J. Phys. A: Math. Theor. 42 335303) we solved a countably infinite family of one-dimensional Schrödinger 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.