# Green function for the particle hopping on a lattice, meaning?

1. Jan 9, 2014

### bulgakov

I am having trouble understanding something that I am sure is very basic. Let's say I have a particle that is hopping on a 1d lattice with a hard wall at x=0 in the presence of some potential - anything, say linear $H_0=F*i$ or Coulomb $H_0=C/i$ where i is the label of the site the particle is on. I treat nearest neighbor hopping as a perturbation. I then calculate the self energy for $i=1$ by adding up all the diagrams that correspond to the particle hopping away from site 1 and then back to it. Using that self-energy I obtain the expression for the Green's function $G_1(w)$ (w is the complex frequency/energy, 1 stands for site 1) and the imaginary part of it gives me the density of states.

Now my question: ahem, what did I just calculate, exactly? For an attractive Coulomb potential this gave me an infinite number of delta functions - bound states, followed by a continuum of scattered states which look right for the Coulomb potential problem on a lattice. Why is it that I get all the eigenvalues of the system from $Im[G_1]$? Naively, I expected to get $E_1$ from $G_1$, $E_2$ from $G_2$, etc. Would I get the same eigenvalues if I calculated $G_2$? (I know I could check by actually doing it, but either way, I clearly don't understand what I am doing even though I seem to be getting correct results).

Any help would be appreciated :)

Last edited: Jan 9, 2014