Poles of GF (time-independent)

From college to graduate school, it took me years to get used to Green's function. I often wonder if it's the same case for others or am I just not too bright. GF is of course an extremely powerful method that takes many years of reading and practice until one is completely comfortable with it. Physics undergraduates probably learned about it---as I did---in a mathematics class, but it is usually unused by physics classes. For example, Griffiths' EM book---standard textbook on the subject---do not use and GF and his QM book only uses it in the scattering theory chapter (for which GF is very useful) at the very end.

My hand-waving understanding of GF is the following: it is the inverse of the Hamiltonian/Linear operator in the following sense:

with the parameter .

Suppose we already know how to diagonalize . It can have both discrete and continuous eigen-kets: and . Then is simply:

One can sandwich other sides whatever complete set you prefer to get the representation in that set, e.g. for space.

Since is hermitian, all its eigenvalues and are real. So has singularities on the real lines. It is otherwise analytic. The poles of therefore gives discrete eigenvalues of . On where belong to the continuous spectrum of , the second term is not well defined since the integrand has a pole. Usually if the eigenstates associated with the continuous spectrum are propagating or extended (do not decay fast), the side limits of as exist but are different on both side. Let

Obviously . We'll express the continuity by a delta-function:

Using the identity

and taking the trace of , we have:

The imaginary part of contains . It is the density of state (DOS) at , :

So there are physical quantities encoded in the Green's function after all. Finally can be expressed by the discontinuity in a Kramers-Kronig-like relation: