Painlevé IV solutions from systems with a harmonic oscillator gapped spectrum
Supersymmetry transformations of order k are applied to the harmonic oscillator for generating potentials V k j whose spectra have a gap of thickness k+1 with respect to the initial spectrum. The system’s extremal states are identified and, since the conditions ensuring that the Hamiltonian has third order ladder operators and thus it is connected with the PIV equation are satisfied, solutions to this equation can be found. An alternative supersymmetry transformation is applied to the harmonic oscillator by adding the levels needed to reproduce the spectrum of V k j, up to a constant energy displacement. The three new extremal states are as well identified and we get the corresponding solutions to the PIV equation. Finally, the PIV solutions found through both transformations are analysed.