Supersymmetry of $\mathcal{PT}$- symmetric tridiagonal Hamiltonians

TL;DR

This paper extends supersymmetry concepts to non-Hermitian, $ ext{PT}$-symmetric tridiagonal Hamiltonians, establishing relations between Hamiltonian pairs, connecting eigenstate polynomials, and applying the formalism to exactly solve a $ ext{PT}$-symmetric Morse oscillator.

Contribution

It introduces a formalism for supersymmetry in non-Hermitian $ ext{PT}$-symmetric Hamiltonians, linking matrix elements and eigenpolynomials, and demonstrates an exact solution for a specific oscillator.

Findings

Established relations between Hamiltonian and partner matrix elements.

Connected eigenstate polynomials via kernel polynomials.

Solved the $ ext{PT}$-symmetric Morse oscillator exactly.

Abstract

We extend the study of supersymmetric tridiagonal Hamiltonians to the case of non-Hermitian Hamiltonians with real or complex conjugate eigenvalues. We find the relation between matrix elements of the non-Hermitian Hamiltonian $H$ and its supersymmetric partner $H^{+}$ in a given basis. Moreover, the orthogonal polynomials in the eigenstate expansion problem attached to $H^{+}$ can be recovered from those polynomials arising from the same problem for $H$ with the help of kernel polynomials. Besides its generality, the developed formalism in this work is a natural home for using the numerically powerful Gauss quadrature techniques in probing the nature of some physical quantities such as the energy spectrum of $\mathcal{PT}$-symmetric complex potentials. Finally, we solve the shifted $\mathcal{PT}$-symmetric Morse oscillator exactly in the tridiagonal representation.