A Surprise in Sum Rules - Modulating Factors

A generic physical situation is considered where Im $\Pi$, the imaginary part of polarization operator (generalized susceptibility), can be measured on a finite interval and the high frequency asymptotics (up to a few orders) of $\Pi$ can be calculated theoretically. In such a case, it is desirable to derive an equivalent form of the Kramers-Kronig dispersion relation, the so-called sum rule, in which both the high-frequency part of Im $\Pi$ in the dispersion integral and the high-order contribution to $\Pi$ are suppressed. We provide a general framework for derivation of such sum rules, without any recourse to an infinite-order differential operator. We derive sum rules for a wide set of weight functions and show that any departure from the $e^{-t}$ behaviour of the weight function in sum rules leads to modulating factors on the theoretical side of sum rules, providing its low frequency regularization. We argue that by including modulating factors one can extend the domain of validity of sum rules further to an intermediate region of frequencies and can account for ``bumps" which were observed numerically on the phenomenological side of sum rules at ``intermediate'' frequencies.