Factorization method and new potentials from the inverted oscillator

TL;DR

This paper applies supersymmetric quantum mechanics to generate new exactly-solvable real potentials from the inverted oscillator, revealing that only complex second-order transformations yield non-singular solutions and exploring the algebraic structure of these Hamiltonians.

Contribution

It introduces a novel application of supersymmetric quantum mechanics to the inverted oscillator, identifying specific complex transformations that produce non-singular potentials.

Findings

Only complex second-order transformations produce non-singular potentials.

New algebraic structures are identified in the resulting Hamiltonians.

The method extends the class of exactly-solvable potentials derived from the inverted oscillator.

Abstract

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.