Properties of shape-invariant tridiagonal Hamiltonians

As is known, a nonnegative-definite Hamiltonian $H$ that has a tridiagonal matrix representation in a basis set allows defining forward (and backward) shift operators that can be used to determine the matrix representation of the supersymmetric partner Hamiltonian $H^{(+)}$ in the same basis. We show that if the Hamiltonian is also shape-invariant, then the matrix elements of the Hamiltonian are related such that the energy spectrum is known in terms of these elements. It is also possible to determine the matrix elements of the hierarchy of supersymmetric partner Hamiltonians. Moreover, we derive the coherent states associated with this type of Hamiltonian and illustrate our results with examples from well-studied shape-invariant Hamiltonians that also have a tridiagonal matrix representation.