Non-isospectral Hamiltonians, intertwining operators and hidden hermiticity

We have recently proposed a strategy to produce, starting from a given Hamiltonian h1h1 and a certain operator x   for which [h1,xx†]=0[h1,xx†]=0 and x†xx†x is invertible, a second Hamiltonian h2h2 with the same eigenvalues as h1h1 and whose eigenvectors are related to those of h1h1 by x†x†. Here we extend this procedure to build up a second Hamiltonian, whose eigenvalues are different from those of h1h1, and whose eigenvectors are still related as before. This new procedure is also extended to crypto-hermitian Hamiltonians.

► Given two operators (h,x)(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.