# Green's function at boundaries

• A
The derivative of the Green's function is:
$$i \dfrac{dG_{A,B}(t)}{dt} =\delta(t) \left< {[A,B]}\right>+G_{[A,H],B}(t)$$
the fourier transform is:
$$\omega G_{A,B}(t)=\left< {[A,B]}\right>+G_{[A,H],B}(\omega)$$
but this would require that the Green's function is 0 for t->inf. Why is that the case? It is clear that it must vanish at t->-inf because of the heaviside function but not at inf.

## Answers and Replies

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

Can you mention the source of these equations. I have never seen such form of the Green's function and would love to know more about it.
Anyways, second equation seems wrong. Instead of
$$\omega G_{A,B}(t)=\left< {[A,B]}\right>+G_{[A,H],B}(\omega)$$
you should have
$$\omega \tilde{G}_{A,B}(\omega)=\left< {[A,B]}\right>+G_{[A,H],B}(\omega)$$
where
$$\tilde{G}_{A,B}(\omega)=\int dt G_{A,B}(t)e^{+i\omega t}$$