# Formal definition of Green function

The formal definition of a Green's function is: $$L(\mathbf{r})G(\mathbf r,\mathbf r^\prime) = \delta(\mathbf r-\mathbf r^\prime), \tag 1$$ 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) ?

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.
• From equation (2) of your answer one gets: $\left[λ_{n} - L(r)\right]\phi_{n}(r)=0$ so the factor $λ_{n} - L(r)$ comes out. But, how to link it to the actual equation (equation 1 of your answer ) for which Green's function is to be defined ? – iknownothing Jul 11 '18 at 11:18
• For instance if you have the Schrodinger equation, and you need to find how an energy eigenfunction responds to a potential. Here the potential would be f(r), the eigenvalue would be $\lambda$ and the eigenfunction would be u(r). – Hari Jul 11 '18 at 13:38