how can i solve $ -ixDG(x,s)-iG(x,s)/2= \delta ( \frac{x}{s}-1) $
i do not know , since it is a first odrder differntial operator, the formal solution i've found would be
$ G(x,s)= \sum_{n} \frac{u_{n}(x)u_{m}(s)}{\lambda _{n}} $
where $ -ixDu_{n}(x)-iu_{n}(x)/2= u_{n} \lambda _{n} $
here G(x,s) is the analogue of the Green function for our operator :)
You have to be specific about the boundary conditions that pick out the Green function that you want. Otherwise the answer is not unique. Let me presume "retarded" boundary conditions, namely $G(x,s) = 0$ for $x<s$. Since the left hand side of the differential equation you are trying to solve is a first order differential operator applied to $G(x,s)$ and the right hand side is a $\delta$-function, you expect the first derivative of $G(x,s)$ to behave like a $\delta$-function near $x=s$, which translates into a jump discontinuity of $G(x,s)$ itself. The size of the jump is isolated by integrating both sides of the differential equation on the interval $[s-\epsilon,s+\epsilon]$ and taking the limit $\epsilon\to 0$. Away from the discontinuity, $G(x,s)$ solves the homogeneous differential equation, which has only one independent solution $C/\sqrt{x}$. The constants $C$ is different on either side of the discontinuity. On the left it is fixed to $C=0$ by the retarded boundary condition. On the right it is fixed by the size of the jump. The answer is \begin{equation} G(x,s) = i \sqrt{s/x}\Theta(x/s-1) , \end{equation} where $\Theta(x)$ is the Heaviside step function. For any other choice of boundary conditions, the corresponding Green function is obtained from the one above by adding some homogeneous solution.
