Variational Principle for Eigenvalue Equations

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Abstract
A variational principle is developed to provide an estimate of an arbitrary functional of the eigen‐functions of a set of eigenvalue equations. It is shown that the variational formalism is equivalent to a functional Taylor series expansion of the desired functional about the trial functions. The relationship of this work to perturbation theory is considered, and it is shown that the formalism can be used to construct higher‐order variational principles, i.e., those for which first‐order errors in the trial functions leads to an nth‐order error in the desired functional. Finally, it is shown that the variational principle of Borowitz and Vassell for estimating off‐diagonal matrix elements, as well as the usual Rayleigh quotient, are special cases of the principle presented here.

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