On a novel supersymmetric connections between harmonic and isotonic oscillators

Beginning from the harmonic oscillator we obtain a one-parameter family of potentials whose spectra coincide with that of the isotonic oscillator. Diverse arguments concerning supersymmetric transformations, factorization procedures and Riccati equations are used in a systematic way. In particular we take advantage of the non-standard factorization of the Schrödinger equation which appears when considering the energy of the first excited level as reference. To be more specific, supersymmetry is neither exact nor spontaneously broken but realizes itself acting on wave functions vanishing at x = 0.