Define a single-particle Green's function as \begin{equation} i\hbar G(xt;x't') = \langle x| e^{-iH(t-t')/\hbar} | x'\rangle. \end{equation} By inserting the completeness relation, we have \begin{equation} i\hbar G(xt;x't') = \sum_n \langle x|n\rangle \langle n| x'\rangle e^{-iE_n(t-t')/\hbar}, \end{equation} where $|n\rangle$ and $E_n$ are the eigenstates and eigenvalue of the Hamiltonian $H$. By using the Fourier transform, the Green's function in the energy domain can be calculated as \begin{equation} \begin{split} G(x,x';E) &= \int_{-\infty}^{\infty} G(xt;x't') e^{iE(t-t')/\hbar} dt\\ &= \frac{1}{i\hbar}\int_{-\infty}^{\infty} \left\{ \sum_n \langle x|n\rangle \langle n| x'\rangle e^{-iE_n(t-t')/\hbar} \right\} e^{iE(t-t')/\hbar} dt\\ &= \frac{1}{i\hbar}\sum_n \langle x|n\rangle \langle n| x'\rangle \int_{-\infty}^\infty e^{i(E-E_n)(t-t')/\hbar} dt\\ &= -2\pi i\sum_n \langle x|n\rangle \langle n| x'\rangle \delta(E-E_n) \end{split} \end{equation} However, from my knowledge, $G(x,x';E)$ is usually defined as \begin{equation} G(x,x';E) = \sum_n\frac{\langle x|n\rangle\langle n| x'\rangle}{E - E_n}. \end{equation} So my question is how can the above two equations be related? Or is there anything wrong with my derivation?

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    $\begingroup$ There are several kinds of Green's functions with different definitions. Your derivation seems correct, so I suspect you're confusing two different kinds of Green's functions up. $\endgroup$ – honey.mustard Jul 29 '19 at 7:59

Both are correct, but are different quantities.

The latter is the (Fourier transform of the) retarded Green's function $G_R$. It is related to your "Green's function" (which really is a kernel) $G$ through

$$ G_R(x,t; x', t') = \frac{1}{i\hbar}\Theta(t-t')G(x,t; x',t') $$

See this wonderful answer about kernels, propagators and Green's functions.

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  • $\begingroup$ Thanks for your reply. First of all, I forget a prefactor $i\hbar$ in my previous definition of the Green's function, which has been added. (Any way, it doesn't matter for the question). Besides, I also read the materials you cited, and agree with you that the problem come from the difference between retarded Green's function and kernel. However, I still cannot quite understand about these two quantities from that answer. Could you provide some other references to this concept? thanks so much. $\endgroup$ – user8036269 Jul 29 '19 at 15:48
  • $\begingroup$ @user8036269 This question and the answer within might shed some light on the matter. $\endgroup$ – Nephente Jul 29 '19 at 19:56

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