Kinetic contribution to the arbitrary order odd frequency moments of the dynamic structure factor

TL;DR

This paper derives an exact expression for the kinetic contribution to odd frequency moments of the dynamic structure factor, applicable to both non-interacting and interacting Fermi gases, validated against known and new results.

Contribution

It introduces a generalized, exact formula for odd frequency moments of the dynamic structure factor, extending previous results to higher orders and interacting systems.

Findings

Validated against known first and third moments

Provided explicit calculations for fifth and seventh moments

Demonstrated applicability to interacting Fermi gases

Abstract

An exact expression is derived for the kinetic contribution to the odd (arbitrary order) frequency moments of the dynamic structure factor via a finite summation that features averages of even (all lower orders) powers of the momentum over the exact momentum distribution. The derivation is carried out for the non-interacting Fermi gas and generalized to the interacting case based on the conjecture that averages over the Fermi distribution can be substituted with averages over the exact distribution. The expression is validated against known results (first, third frequency moments) and new explicit calculations (fifth, seventh frequency moments).