Upper and lower bounds to quantum-mechanical sum rules

It is pointed out that when a set of sum rules S ( k ) are composed into an array S with elements S ij = S ( i + j ), the resulting matrix S has no negative eigenvalues. This property often permits one to give rigorous upper and/or lower bounds to the true value of a particular sum rule when values of other sum rules are known. In certain cases the bounds are identical with those obtained recently by Gordon using a generalized theory of Gaussian integration. The technique is illustrated by an application to oscillator strength sum rules in the ground state of the hydrogen atom and the negative hydrogen ion, and finally it is shown how any available information about individual oscillator strengths can be applied to further improve the bounds.