Supersymmetric partners of the harmonic oscillator with an infinite potential barrier

Supersymmetry transformations of first-and second-order are used to generate Hamiltonians with known spectra departing from the harmonic oscillator with an infinite potential barrier. Also studied is the way in which the eigenfunctions of the initial Hamiltonian are transformed. The first-and certain second-order supersymmetric partners of the initial Hamiltonian possess third-order differential ladder operators. Since systems with this kind of operators are linked with the Painleve IV equation, several solutions of this nonlinear second-order differential equation will be simply found.