The Rayleigh-Ritz minimization principle is generalized to ensembles of unequally weighted states. Given the M lowest eigenvalues $E_1$≤$E_2$≤...≤$E_M$ of a Hamiltonian H, and given M real numbers $w_1$≥$w_2$≥...≥$w_M$>0, an upper bound for the weighted sum $w_1$$E_1$ +$w_2$$E_2$+...+$w_M$$E_M$ is established. Particular cases are the ground-state Rayleigh-Ritz principle (M=1) and the variational principle for equiensembles ($w_1$=$w_2$=...=$w_M$). Applications of the generalized principle are discussed.

• Received 21 October 1987

DOI:https://doi.org/10.1103/PhysRevA.37.2805