Unusual isospectral factorizations of shape invariant Hamiltonians with Scarf II potential

TL;DR

This paper explores novel isospectral factorizations of shape invariant Hamiltonians with the Scarf II potential, revealing complex hierarchies with real spectra and their algebraic structures.

Contribution

It introduces two classes of factorizations for the Scarf II Hamiltonian, including a complex hierarchy with real spectra, expanding the understanding of potential algebra structures.

Findings

Complex Hamiltonians are not PT-symmetric but have real spectra.

The potential algebra is composed of two $su(1,1)$ algebras.

Real and complex hierarchies are isospectral to the original Hamiltonian.

Abstract

In this paper, we search the factorizations of the shape invariant Hamiltonians with Scarf II potential. We find two classes; one of them is the standard real factorization which leads us to a real hierarchy of potentials and their energy levels; the other one is complex and it leads us naturally to a hierarchy of complex Hamiltonians. We will show some properties of these complex Hamiltonians: they are not parity-time (or PT) symmetric, but their spectrum is real and isospectral to the Scarf II real Hamiltonian hierarchy. The algebras for real and complex shift operators (also called potential algebras) are computed; they consist of $su(1,1)$ for each of them and the total potential algebra including both hierarchies is the direct sum $su(1,1)\oplus su(1,1)$.