  A single band optical sum rule derived by Kubo can reveal a novel kind of
superconducting state. It relies, however, on a knowledge of the single band
contribution from zero to infinite frequency. A number of experiments over the
past five years have used this sum rule; their data has been interpreted in
support of 'kinetic energy-driven superconductivity'. However, because of the
presence of unwanted interband optical spectral weight, they necessarily have
to truncate their sum at a finite frequency. This work examines theoretical
models where the impact of this truncation can be examined first in the normal
state, and then in the superconducting state. The latter case is particularly
important as previous considerations attributed the observed anomalous
temperature dependence as an artifact of a non-infinite cutoff frequency. We
find that this is in fact not the case, and that the sign of the corrections
from the use of a non-infinite cutoff is such that the observed temperature
dependence is even more anomalous when proper account is taken of the cutoff.
On the other hand, in these same models, we find that the strong observed
temperature dependence in the normal state can be attributed to the effect of a
non-infinite cutoff frequency.
