friends.
My question is the following:
So I need to find a Green function (an inverse operator kernel) for an operator, which is 'kind of similar' to a transport operator.
$$ L = \frac{\partial}{\partial x_1} + \frac{\partial}{\partial x_2} - i $$
writing an equation, I get:
$$ Lg(x) = \delta(x) $$
Using Fourier transform, the equation is turned into an algebraic one:
$$ (-iy_1 - iy_2 - i)h(y) = 1 $$
I cannot solve it further.
Namely, there are three problems:
0) $ 1/(-iy_1 - iy_2 - i) $ doesn't belong to the Schwarz space, which means that I need to fins some regularization of it. How? If the function only had a point support -- I could have done something like a PV integral, but here I am puzzled.
1)Since $ 1/(-iy_1 - iy_2 - i) $ doesn't belong to Schwarz space, the homogenious equation has non-trivial roots.
$ (-iy_1 - iy_2 - i) h(y) = 0 $
Again, I don't know how to find them. They could have been just deltas at various points of the domain, where the inverse doesn't exist, but here there is a whole line.
2) I don't know how to find an inverse Fourier of $ 1/ (-iy_1 - iy_2 - i) $, since it has a singularity on the real line and thus I don't know how to apply residue theory to them.
Any suggestions?
Actually, I have already asked for help, and received the following hint:
![http://www.codecogs.com/gif.latex?\hat{F}^{-1}_{k_1}\left[%20\frac{1}{-ik_1%20-%20ik_2%20-i}\right]%20=%20e^{-i(1+k_2)x_1}\left(\frac{1}{2}-\theta(x_1)\right)](https://i.stack.imgur.com/PYt1a.gif)
But to be honest, I don't know how to use it :-( I guess, it should help me take the inverse Fourier, but I am yet far from there.
Any suggestions (or links to book chapters) would be greatly appreciated.
Actually, I know a solution for the case when we have 1 instead of $i$. Then the operator is exactly a transport equation, and It is solved by plain dividing by $1/(−iy1−iy2−1)$ , but here it is not the case.
Also, I kind of know something about the homogeneous part.... just by plugging in $e^{ix_1 + ix_2}$