Sum rules for nonlinear plasma response functions

The asymptotic high-frequency ($\omega \to \infty$) and static ($\omega =0\mathrm{}$) behavior of the quadratic longitudinal polarizability, combined with the Kramers-Kronig relations, leads to sum rules analogous to the linear "conductivity" and "compressibility" rules. Further, Kubo-type sum rules are derived by taking frequency moments of the nonlinear fluctuation-dissipation theorem; these lead to the external quadratic polarizability summing to static linear response functions.