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

M. W. AlMasri

Published 2020-06-16Version 1

The notion of pseudo-Hermiticity is introduced in the context of supersymmetric tridiagonal Hamiltonians. Our formalism works for Hermitian and non-Hermitian Hamiltonians with real or complex-conjugate pair eigenvalues . We find the relation between matrix elements of the 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. We apply our formalism to the case of shifted $\mathscr{PT}$-symmetric Morse oscillator.