Let Assume I have a differential inhomogeneous equation
$$(L-\lambda ) u(x) = f(x)$$
Where L is the Hermitian linear differential operator. My problem is that I want to connect the function u(x) with the green function.
I have seen an equation which can be written for the above equation
$$u(x) \int G_\lambda (x,y) f(y) dy$$ and therefore $$ (L-\lambda) G_\lambda (x,y) = \delta(x-y)$$
Could you please elaborate the connections from the first equations to the rest of the two equations more explicitly?
Denote the operator $L-\lambda$ by $\tilde L$. To solve $\tilde Lu = f$, perhaps the most intuitive way is to find the inverse operator $(\tilde L)^{-1}$. We represent $(\tilde L)^{-1}$ as an integral kernel $G(x,y)$ satisfying $$ \tilde L_x G(x,y) = \delta(x-y),$$ where the subscript on $\tilde L_x$ means $\tilde L$ acts on the first argument of $G$. With this representation in mind, the solution to $\tilde Lu = (L-\lambda)u =f$ can be written as $$ u(x) = \int G(x,y)f(y)\, dy, $$ since $$ \tilde Lu(x) = \int \tilde L_xG(x,y)f(y)\, dy = \int \delta(x-y)f(y)\, dy = f(x). $$