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Addition of Complex Term in Lippmann Schwinger Equation

  1. Jan 31, 2012 #1
    Hello PF People,

    This is probably a very simple question but I don't really get it.

    In Lippmann Schwinger Equation we add an infinitesimal term to the denominator in order to avoid singularity for when E is an eigenvalue of [itex]\hat{H}_0[/itex]. This is fine, but why it has to be a complex number? Adding a real constant will also save denominator from becoming singular. That confuses me.

    Thanks in advance!
     
  2. jcsd
  3. Feb 1, 2012 #2
    Nobody? It seemed to be a good question.

    My point of view: I guess the reason is to avoid more singularities, as could be the case of continuous spectra. Knowing that the limit [itex] \epsilon \rightarrow 0 [/itex] would be the same in all directions (imaginary, real or "mixed complex")

    There are no complex energies in the spectrum, so, that should be "safer".
     
  4. Feb 1, 2012 #3

    Bill_K

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    Science Advisor

    The eigenvalue spectrum of H0 is continuous, all real numbers E0 > 0. In order to give meaning to the inverse operator (E - H0)-1 it is necessary to avoid the entire positive real axis. The fact that we use +iε rather than -iε has to do with the outgoing boundary conditions.
     
  5. Feb 1, 2012 #4
    Ahh, i forgot that was indeed the free hamiltonian, then seems obvious. I have my scattering theory rusty, damn it.
     
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