We have an operator of green function which can be written as $\Big(-i \frac{\partial}{\partial x} -a \Big)^2$ within the periodic boundary conditions, where $x,y \epsilon ]-\pi, \pi].$
Using that operation I have derived a GREEN function which is $$G(x,y) = \frac{1}{2} e^{i(x-y)a}\Big(- |x-y|- i(x-y) \cot \pi a+\frac{\pi}{\sin^2 \pi a} \Big)$$
Summary : I have to show the Green function like $G(-\pi , y) = G(\pi, y)$
I understand how the derivation works and it's pretty long. If anyone asks me how can I verify that Green function operator directly with that operator?** ****I didn't understand what is the activity of** active crease? and why (x-y) should be zero for for Green function?
What I understand is that, I have to verify the periodicity of G (x,y) over the boundary conditions they have given. What terms in the Green function should verify the periodicity in Fourier Space for the operator I mentioned first?
Edit:
After I apply the operator on G a, I got $(-i \partial_x - a)^2 G = \delta$ and using this notation, I found the coefficients of $g_n$.
$G(x,y) = \sum_{n=-\infty} ^{\infty} g_n e^{n(x-y)}$
Since I know $g_n$ I understand how I got the green function expression $$G(x,y) = \frac{1}{2} e^{i(x-y)a}\Big(- |x-y|- i(x-y) \cot \pi a+\frac{\pi}{\sin^2 \pi a} \Big)$$