Does random phase approximation (RPA) response function obey Kramers-Kronig relations? Consider the screened Coulomb interaction in electron liquid, which in the random phase approximation (RPA) takes the form
$$
V(q,\omega)=\frac{v(q)}{1-v(q)\Pi(q,\omega)},
$$
where $v(q)$ is the unscreened Coulomb interaction, $\Pi(q,\omega)$ is the electron gas polarizability.
It is known the exact screened interaction $V_\mathrm{exact}(q,\omega)$ obeys the Kramers-Kronig relations
$$
\mathrm{Re}\,V_\mathrm{exact}(q,\omega)=v(q)-\frac1\pi\mathcal{P}\int d\omega'\frac{\mathrm{Im}\,V_\mathrm{exact}(q,\omega')}{\omega-\omega'},
$$
$$
\mathrm{Im}\,V_\mathrm{exact}(q,\omega)=\frac1\pi\mathcal{P}\int d\omega'\frac{\mathrm{Re}\,V_\mathrm{exact}(q,\omega')-v(q)}{\omega-\omega'}.
$$
The polarizability $\Pi(q,\omega)$, being a retarded response function, also obeys similar relations (although without $v(q)$ in the right hand sides).
Does the RPA interaction $V(q,\omega)$ obey the Kramers-Kronig relations? If yes, how it can be proved?
 A: Yes. The RPA response function $V(\omega)$ still obeys the Kramers-Kronig (KK) relation, as long as the polarization function $\Pi(\omega)$ obeys the KK relation. The key point is to show that all the higher-order poles that appear in the RPA expansion can be reduced to first-order using the KK relation of $\Pi(\omega)$, such that they do not cause a problem.
The KK relation of the polarization function implies that we can express $\Pi(\omega)$ as
$$\Pi(\omega)=\int\frac{\mathrm{d}\omega'}{2\pi}\frac{A_\Pi(\omega')}{\omega-\omega'+\mathrm{i}0_+},$$
where $A_\Pi(\omega)\equiv -2\Im\Pi(\omega)$ is the spectral function of the polarization. Then let us focus on the term $v\Pi(\omega)v\Pi(\omega)v$ in the RPA expansion. We want to show that its sencond-order poles actually can be resolved by first-order poles and hence not problematic. To see this, we start from
$$\begin{split}\Pi(\omega)^2&=\int\frac{\mathrm{d}\omega_1}{2\pi}\frac{\mathrm{d}\omega_2}{2\pi}\frac{A_\Pi(\omega_1)}{\omega-\omega_1+\mathrm{i}0_+}\frac{A_\Pi(\omega_2)}{\omega-\omega_2+\mathrm{i}0_+}\\
&=\int\frac{\mathrm{d}\omega_1}{2\pi}\frac{\mathrm{d}\omega_2}{2\pi}\Big(\frac{1}{\omega-\omega_1+\mathrm{i}0_+}-\frac{1}{\omega-\omega_2+\mathrm{i}0_+}\Big)\frac{A_\Pi(\omega_1)A_\Pi(\omega_2)}{\omega_1-\omega_2}\end{split}.$$
Then we can define a new spectral function
$$A_\Pi^{(2)}(\omega)=2\int\frac{\mathrm{d}\omega'}{2\pi}\frac{A_\Pi(\omega)A_\Pi(\omega')}{\omega-\omega'}=2A_\Pi(\omega)\Re\Pi(\omega)=-4\Im\Pi(\omega)\Re\Pi(\omega),$$
which resolves the pole $(\omega-\omega')^{-1}$ in the integrant by the KK relation of $\Pi(\omega)$, hence reducing the total order of the poles by one. The spectral function $A_\Pi^{(2)}$ will be as analytic as $\Pi(\omega)$, which is exactly the spectral function of $\Pi(\omega)^2$ with out sencond-order poles:
$$\Pi(\omega)^2=\int\frac{\mathrm{d}\omega'}{2\pi}\frac{A^{(2)}_\Pi(\omega')}{\omega-\omega'+\mathrm{i}0_+}.$$
Following the above approach, it is straight forward to show that all the higher-order terms in the RPA expansion has the spectral resolution of the same form in terms of the first-order poles only.
$$\Pi(\omega)^n=\int\frac{\mathrm{d}\omega'}{2\pi}\frac{A^{(n)}_\Pi(\omega')}{\omega-\omega'+\mathrm{i}0_+}.$$
All the higher-order spectral functions $A_\Pi^{(n)}(\omega)$ can be express as a polynomial of $\Re\Pi(\omega)$ and $\Im\Pi(\omega)$. So as long as $\Pi(\omega)$ obeys the KK relation, all terms in the RPA expansion will also obeys the KK relation, as well as the RPA response function $V(\omega)$.
