2024-11-1020210217-732310.1142/S02177323215024242-s2.0-85120563251http://dx.doi.org/10.1142/S0217732321502424https://hdl.handle.net/20.500.14288/16482We 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 PT-symmetric complex potentials. Finally, we solve the shifted PT-symmetric Morse oscillator exactly in the tridiagonal representation.AstronomyAstrophysicsPhysics, nuclearPhysics, particles and fieldsPhysics, mathematicalJournal Article1793-6632727114500004Q211039