Supersymmetry of PT-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 PT-symmetric complex potentials. Finally, we solve the shifted PT-symmetric Morse oscillator exactly in the tridiagonal representation.