Green's function at boundaries

[mupsi]

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.

 

[Ravi Mohan]

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
$$\tilde{G}_{A,B}(\omega)=\int dt G_{A,B}(t)e^{+i\omega t}$$