Key points are not available for this paper at this time.
The Rayleigh-Ritz minimization principle is generalized to ensembles of unequally weighted states. Given the M lowest eigenvalues E₁E₂. . . E₌ of a Hamiltonian H, and given M real numbers w₁w₂. . . w₌>0, an upper bound for the weighted sum w₁E₁ +w₂E₂+. . . +w₌E₌ is established. Particular cases are the ground-state Rayleigh-Ritz principle (M=1) and the variational principle for equiensembles (w₁=w₂=. . . =w₌). Applications of the generalized principle are discussed.
Gross et al. (1988) studied this question.