I want to find the general form of the Green's function which satisfies the following equation
$\int^\infty_{-\infty} dy \, G^*(y - x) G(y - x') = \delta(x-x')$.
Is there a general method in which I could find the specific form of $G$ which satisfies this? I have been trying with the following form for G:
$G(y-x) = \frac{e^{- i k (y - x)}}{(y - x)}$, which represents a propagation of a wave from source $x$ to the point $y$ on the detector plane.
What is $G^{\ast }$? If $G(y-x)=<y|K|x>$ with some operator $K$ then \begin{eqnarray*} \delta (x-x^{\prime }) &=&<x|K^{-1}K|x^{\prime }>=\int dy<x|K^{-1}|y><y|K|x^{\prime }> \\ &=&\int dy<x|K^{-1}|y>G(y-x^{\prime }) \end{eqnarray*} Thus, if $K^{-1}=K^{\ast }$ then your relation follows. This is the case for unitary $K$.