Factorization method and new potentials from the inverted oscillator

•We apply supersymmetric quantum mechanics to the inverted oscillator potential.•The complex second-order transformations allow us to build new non-singular potentials.•The algebraic structure of the initial and final potentials is analyzed.•The initial potential is described by a complex-deformed Heisenberg–Weyl algebra.•The final potentials are described by polynomial Heisenberg algebras.

In this article we will apply the first- and second-order supersymmetric quantum mechanics to obtain new exactly-solvable real potentials departing from the inverted oscillator potential. This system has some special properties; in particular, only very specific second-order transformations produce non-singular real potentials. It will be shown that these transformations turn out to be the so-called complex ones. Moreover, we will study the factorization method applied to the inverted oscillator and the algebraic structure of the new Hamiltonians.