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Hamiltonian and Green functions.

  1. Apr 5, 2007 #1

    tpm

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    Let be the 1-D Hamiltonian:

    [tex] \hat H = -\hat D ^{2} + V(\hat x) [/tex] (1)

    and its associated 'Green function' so:

    [tex] -D^{2} G(x,s)+V(x)G(x,s)=\delta (x-s) [/tex] (2)

    then my question is if there is a relationship between:

    [tex] Tr[exp(-a\hat H ) ] [/tex] and [tex] Det[1-aG] [/tex] (3)

    where a >0 and 'G' is the Kernel or Green function (due to the relationship between ODE's and Integral equation..

    In case (3) is not exact i would like to know if there is any relationship between the trace of a Hamiltonian and the Determinant of its Fredholm Operator (in this case the Green function or propagator)
     
  2. jcsd
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