Sum rules for the frequency spectrum of linear magnetic chains

Exact sum rules are obtained for the nearest-neighbor Heisenberg antiferromagnet and $X Y$ ferromagnet in one dimension. In the Heisenberg case the sum rules may be used to show that most of the spectral weight of the spin correlation function at small $k$ and $T=0\mathrm{}$ is concentrated near the frequency of the "des Cloizeaux-Pearson" states. In the $X Y$ case, the corresponding "Schultz-Lieb-Mattis" states only carry a negligible portion of the weight at small $k$.