On the application of the Rayleigh-Ritz method to a projected Hamiltonian

We apply the well known Rayleigh-Ritz method (RRM) to the projection of a Hamiltonian operator chosen recently for the extension of the Rayleigh-Ritz variational principle to ensemble states. By means of a toy model we show that the RRM eigenvalues approach to those of the projected Hamiltonian from below in most cases but a few ones. We also discuss the effect of an energy shift and the projection of the identity operator.