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There is a argument about response function: according to the Kramers-Kronig relation$$G(\omega)=\int_{-\infty}^{+\infty}d\omega' \frac{A(\omega')}{\omega+i0_+-\omega'}$$ response function will change sign at the energy scale in which there exists “bump” in the spectral function $A(\omega)$. I am confuse of the origin of this arguments since I cannot find it from the expresion directly.

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I'm assuming you mean that the real part of the Green function will change its sign. The imaginary part of the Green function is related to the spectral density and therefore it is always negative (from the realtion $-\rm{Im}\{G^r(\omega)\} = \pi A(\omega)$).

To see what you are looking for you can examine it in the following manner: $$ \rm{Re}\{G^r(\omega)\} = \int\!d\omega' \frac{\omega-\omega'}{(\omega-\omega')^2+0_+^2}A(\omega')$$ and since $A(\omega')$ and the denominator are always positive, the sign of the integral is determined by the numerator. So if $A(\omega)$ has a "bump" at some $\omega_0$ (that is - it is concentrated around that frequency), then for $\omega>\omega_0$ the integral will be mostly positive, while for $\omega < \omega_0$ it will be mostly negative.

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