Non-isospectral Hamiltonians, intertwining operators and hidden hermiticity

We have recently proposed a strategy to produce, starting from a given Hamiltonian h1 and a certain operator x for which [h1,xx†]=0 and x†x is invertible, a second Hamiltonian h2 with the same eigenvalues as h1 and whose eigenvectors are related to those of h1 by x†. Here we extend this procedure to build up a second Hamiltonian, whose eigenvalues are different from those of h1, and whose eigenvectors are still related as before. This new procedure is also extended to crypto-hermitian Hamiltonians. -- Highlights: ► Given two operators (h,x) we define H whose eigensystem is deduced from that of h. ► We extend this procedure to a crypto-hermitian Hamiltonians. ► We discuss the role of intertwining operators, bounded or not, in this construction.