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

  1. Jan 9, 2014 #1
    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
  2. jcsd
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