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arXiv · 2006.09544

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

Mohammad Walid AlMasri
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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.

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Mohammad Walid AlMasri. 2020-06-16. Supersymmetry of $\mathcal{PT}$- symmetric tridiagonal Hamiltonians. https://doi.org/10.1142/s0217732321502424

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