Sufficiency of the anticorrelation identities

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Abstract
It is shown that the satisfaction of any two of the four anticorrelation identities recently derived by Ullah is a sufficient condition for the wave function involved to be an exact eigenfunction. Furthermore, a particular linear combination is equivalent to the variance, whose vanishing is in itself a sufficient condition for an exact eigenfunction. Further positive semidefinite expressions, whose vanishing is a necessary and sufficient condition for an exact eigenfunction, are suggested. These and related properties are illustrated by means of two simple examples which exhibit some of the underlying features of applying anticorrelation identities to the optimization of trial functions.