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In the book The theory of quantum liquids by Pines and Nozzieres, I have trouble understanding how one goes from formula 2.58 to formula 2.62 and 2.63 on page 99.

So,one defines the response function (Eq. 2.58) as

$\chi (q,\omega) = \int_{0}^{\infty} d\omega'S(q,\omega')\bigg(\frac{1}{\omega-\omega' +i\eta}-\frac{1}{\omega+\omega' +i\eta}\bigg) \\ $
where

$ S(q,\omega) = \sum_m |\langle m|F|0\rangle|^2 \delta(\omega - \omega_{m0}) $

One splits $ \chi $ as

$\chi = \chi' + i\chi'' $

And one uses the following relation,

$\lim_{\eta \rightarrow 0} \frac{1}{x-a+i\eta}=P\frac{1}{x-a}-i\pi\delta(x-a)$

and one gets Eq 2.62 and Eq 2.63

$ \chi'= \int_{0}^{\infty} d\omega'S(q,\omega') P\big (\frac{2\omega'}{\omega^2-\omega'^2} \big) $

$ \chi''= -\pi(S(q,\omega)-S(q,-\omega)) $

Thanks!

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  • $\begingroup$ $S(q,\omega)$ is clearly real from the formula you have written, so the 2 equations you are aiming for simply come from direct substitution of the Sokhotski-Plemelj formula you quoted into the equation for $\chi$ and taking real and imaginary parts. Can you be a bit more explicit about what is not clear? $\endgroup$ – By Symmetry Aug 18 '17 at 9:34
  • $\begingroup$ Yes. So,we have $ \sum_m |\langle m|F|0\rangle|^2 \big ( \int_{0}^{\infty} d\omega'\frac{\delta(\omega' - \omega_{m0}) }{\omega-\omega' +i\eta}- \int_{0}^{\infty} d\omega'\frac{\delta(\omega' - \omega_{m0}) }{\omega+\omega' +i\eta} \big ) $. The Sokhotski-Plemelj applies when we have a function on top of the fraction..there we have a distribution..so that's why it does not make sense to me.. $\endgroup$ – Small Pole Aug 18 '17 at 9:47
  • $\begingroup$ If you look at formula 6.2 in here :physics.drexel.edu/~tim/open/mas/node10.html you would see would get an integral of two delta function which doesn't make sens cause you cannot multiply two deltas $\endgroup$ – Small Pole Aug 18 '17 at 10:15

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