Supersymmetry of tridiagonal Hamiltonians

Abstract

A positive semi-definite Hamiltonian H that has a tridiagonal matrix representation in a basis set, allows a definition of forward- and backward-shift operators that can be used to define the matrix representation of its supersymmetric partner Hamiltonian H( + ) with respect to the same basis. We find explicit relationships connecting the matrix elements of both Hamiltonians. We present a method to obtain the orthogonal polynomials in the eigenstate expansion problem attached to H( + ) starting from those polynomials arising in the same problem for H. This connection is established by using the notion of kernel polynomials. We apply the obtained results to two known solvable models with different kinds of spectrum.