Supersymmetry of the Morse oscillator

TL;DR

This paper explores the supersymmetry properties of the Morse oscillator, demonstrating that its partner Hamiltonian maintains a tridiagonal matrix form and relating their associated polynomials, extending the formalism to mixed spectra.

Contribution

It applies supersymmetry formalism to the Morse oscillator, showing the partner Hamiltonian's matrix form and polynomial relations, including cases with mixed spectra.

Findings

Partner Hamiltonian also has a tridiagonal matrix representation.

Eigenstate polynomials of the partner are kernel polynomials of the original.

Formalism extends to Hamiltonians with mixed discrete and continuous spectra.

Abstract

While dealing in [1] with the supersymmetry of a tridiagonal Hamiltonian H, we have proved that its partner Hamiltonian H(+) also have a tridiagonal matrix representation in the same basis and that the polynomials associated with the eigenstates expansion of H(+) are precisely the kernel polynomials of those associated with H. This formalism is here applied to the case of the Morse oscillator which may have a finite discrete energy spectrum in addition to a continuous one. This completes the treatment of tridiagonal Hamiltonians with pure continuous energy spectrum, a pure discrete one, or a spectrum of mixed discrete and continous parts.