I am trying to learn about Green's functions as part of my graduate studies and have a rather basic question about them:
In my maths textbooks and a lot of places online, the basic Greens function G for a linear differential operator L is stated as
$$ L G = \delta (x-x') $$
which is all well and good. I am now reading Economou's text on GF in Quantum Physics where he goes to define Green's functions as solutions of inhomogenous DE of the type:
$$ [z - L(r)]G(r,r';z) = \delta (r-r') $$
Where $z = \lambda + is$ and L is a time independent, linear, hermitian differential operator that has eigenfunctions $\phi_n (r)$
$$ L(r) \phi_n (r) = \lambda_n \phi_n (r)$$
Where these $\lambda_n$ are the eigenvalues of L. Where does this z come from in the second equation and what is the link between this and the first one?
Edit: see my post below for a new couple of questions.