Timeline for Where are the poles of the one-particle Green's function located in the complex plane?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 21, 2018 at 4:37 | comment | added | Ian | Let us continue this discussion in chat. | |
| Sep 21, 2018 at 4:30 | comment | added | Ruben Verresen | @Ian I am not sure what you mean by eigenvalue of the resolvent... The poles of the resolvent are the physically meaningful quantity. I thought you were asking whether the resolvent has a pole at $z= \pm i \Gamma$. I don't think it makes sense to ask about eigenvalues of the resolvent; how would you even define that? $\hat G(z) |\psi_z\rangle = \lambda_z |\psi_z\rangle$?... Seems pretty meaningless to me! | |
| Sep 21, 2018 at 4:21 | comment | added | Ian | My (new) knowledge is that for an operator H, the resolvent of H can in general have additional poles that aren't the eigenvalues of H. My question though is whether these poles are necessarily eigenvalues of the resolvent! I'll keep digging. | |
| Sep 21, 2018 at 3:27 | comment | added | Ruben Verresen | @Ian Good point! I'm stumped; a fun question to think about. | |
| Sep 21, 2018 at 3:06 | comment | added | Ian | If you define the resolvent in that way, it's not clear to me how the poles appear without the c_n. Any comment? | |
| Sep 21, 2018 at 2:24 | comment | added | Ruben Verresen | @Ian sure, I don't see what would prevent one from repeating the above argument for $\hat G (z) = (z-\hat H)^{-1}$. | |
| Sep 21, 2018 at 2:01 | comment | added | Ian | This makes a lot of sense. Let me push the question a step further. In the finite n limit, the poles are located at the eigenvalues of the Hamiltonian, such that the eigenvalues of the resolvent are 1/(z-E_n). In your example, is the pole 1/(z +/- i*Gamma) an eigenvalue of the resolvent? | |
| Sep 18, 2018 at 7:34 | history | edited | Ruben Verresen | CC BY-SA 4.0 |
added 68 characters in body
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| Sep 17, 2018 at 22:26 | history | answered | Ruben Verresen | CC BY-SA 4.0 |