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## Main Question or Discussion Point

Hi all! I am currently working on a problem that involves calculation of Green functions. I am having serious doubts on some part of the calculation, as I am getting a GF that has positive values of its imaginary part. According to Lehman's rule, the retarded GF can be decomposed by the spectrum and this decomposition is positive definite. Moreover, it represents a density of states. I know this for LOCAL green functions, but is this also true if I calculate G_ret( 0, 1 ) that is, non-diagonal elements of the retarded GF matrix?

In summary: Terms G_ret(i,i) have Im( G_ret(i,i) ) < 0, but I am having terms with:

[itex] Im[G_{ij}(\omega)] > 0 [/itex]

for [tex] i\neq j [/tex], is that possible? Thanks!

EDIT: I found out that in general we can write the retarded Green's function as:

[itex] G_{ij}^{ret}(\omega) = \sum_{n}\frac{\langle i |A |n\rangle\langle n| B^{\dagger}|j\rangle}{\omega - \varepsilon_{n} + i\eta} [/itex]

for any basis of states satisfying the closure relation. I guess the positivity of the (-) imaginary part is only satisfied when [itex]i=j[/itex], but if I am wrong here please correct me!

In summary: Terms G_ret(i,i) have Im( G_ret(i,i) ) < 0, but I am having terms with:

[itex] Im[G_{ij}(\omega)] > 0 [/itex]

for [tex] i\neq j [/tex], is that possible? Thanks!

EDIT: I found out that in general we can write the retarded Green's function as:

[itex] G_{ij}^{ret}(\omega) = \sum_{n}\frac{\langle i |A |n\rangle\langle n| B^{\dagger}|j\rangle}{\omega - \varepsilon_{n} + i\eta} [/itex]

for any basis of states satisfying the closure relation. I guess the positivity of the (-) imaginary part is only satisfied when [itex]i=j[/itex], but if I am wrong here please correct me!

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