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

TL;DR

This paper explores the application of the Rayleigh-Ritz method to a projected Hamiltonian, demonstrating its eigenvalue behavior and effects of energy shifts through a toy model.

Contribution

It extends the Rayleigh-Ritz variational principle to ensemble states using a projected Hamiltonian, analyzing eigenvalue convergence and methodological effects.

Findings

Eigenvalues approach from below in most cases

Energy shift influences eigenvalue estimates

Projection of the identity operator affects results

Abstract

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.