Formal definition of Green function

Question

The formal definition of a Green's function is: \begin{equation} L(\mathbf{r})G(\mathbf r,\mathbf r^\prime) = \delta(\mathbf r-\mathbf r^\prime), \tag 1 \end{equation} where L is a time linear differential operator.

Now I am reading the book by E. L. Economou's book titled "Green's Functions in Quantum Physics" which gives the definition as, $$ [z - L(\mathbf r)]G(\mathbf r,\mathbf r^\prime;z) = \delta (\mathbf r-\mathbf r^\prime),\tag 2 $$ where $z = \lambda + is$ and L is a time independent, linear, hermitian differential operator that has eigenfunctions $\phi_n (r)$ $$ L(\mathbf r) \phi_n (\mathbf r) = \lambda_n \phi_n (\mathbf r),\tag 3$$ where $\lambda_n$ are the eigenvalues of L.

Surely, Eq. (2) can be considered as the Green function equation for the differential equation, $$ [z - L(\mathbf r)]u(\mathbf r) = f(\mathbf r),\tag 4 $$ such that we can get, $$ u(\mathbf r) = \int f(\mathbf r^\prime) G(\mathbf r,\mathbf r^\prime;z) dr^\prime.\tag 5 $$ Now I am confused!

Why he selected such a form instead of the simpler form in Eq. (1) ? What are the advantages ?

What is really z here ? It is given as a complex variable?

What are intermediate steps that are missing in going from Eq. (1) to (2) ?

Answer 1

For the differential equation $$[z−L(r)]u(r)=f(r), $$ given that the eigenvalues are $ \lambda $ as per $$ L(r)ϕ_{n}(r)=λ_{n}ϕ_{n}(r), $$ the equation for the Green's function would be $$ [λ_{n}−L(r)]G(r,r′;z)=δ(r−r′). $$ The given expression matches this except for the extra complex factor of $ \iota s $. The mathematical justification for this is to shift the poles of the Green's Function to the complex plane. Without this complex value the Green's function would be of the form $ \frac{1}{(λ-L)^{-1}}$ which has poles at the eigenvalues. The inclusion of this complex factor avoids this inconvenience by shifting the poles to the complex plane.

This is quite a common trick in physics and can be seen in a lot of places.