Supersymmetry of $\mathcal{P}\mathcal{T}$- symmetric tridiagonal Hamiltonians

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 $\mathcal{P}\mathcal{T}$-symmetric complex potentials. Finally, we solve the shifted $\mathcal{P}\mathcal{T}$-symmetric Morse oscillator exactly in the tridiagonal representation.

23 pages, 3 figures

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Cited by in corpus (2)

• Integral Transforms and $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians
• Parity-Time Symmetric Spin-1/2 Richardson-Gaudin Models