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I would like to understand how to arrive at the series in equation (36) in this paper https://arxiv.org/abs/cond-mat/0506438, specifically $$\Sigma(E) = V+VG'_0(E)V+VG'_0(E)VG'_0(E)V$$ where $G'_0(E)$ is the free propagator when operating on excited states and 0 when operating on the ground state.

I'd imagine you can derive it from the Dyson equation $$\Sigma = G_0^{-1}-G^{-1},$$ but I don't understand why the Dyson equation doesn't just give $\Sigma = V$.

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  • $\begingroup$ and what is $V$? $\endgroup$ – AccidentalFourierTransform Sep 13 '19 at 22:33
  • $\begingroup$ It's the interaction part of the hamiltonian that's taken as the perturbation. $\endgroup$ – Daniel Sep 14 '19 at 14:12

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