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I came across the following differential equation when considering some direct scattering problems: $$ N'_x(x,z)=G(x,z)N(x,z) $$

where $N(x,z)$ is a $2\times2$ complex matrix with variables $x$ and $z$, $N'_x$ is its derivative with respect to $x$, $G$ is the matrix given by $$\begin{pmatrix} 0 & e^{-izx}u(x)\\\\ e^{izx}\bar{u}(x)& 0 \end{pmatrix} $$ where $u(x)\in L^1(\mathbb{R})\cap L^2(\mathbb{R})$. The boundary condition is $N(\infty, z)=1$.My questions are:Firstly, if I take $z$ to be a real number $s$, does there exist a unique continuous solution of this differential equation? Why or why not? Since general ODE theory requires a better regularity properties on the coefficients, does the theory still hold for the not-that-good coefficients? Secondly, does the solution has a well defined limit as $x\to-\infty$? Why or why not? I would appreciate it if anyone could help me with my questions!

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"General ODE theory" should include Carathéodory's existence theorem en.wikipedia.org/wiki/Carath%C3%A9odory%27s_existence_theorem So existence of an absolutely continuous solution is known. The only issue would be uniqueness. Now since your equation is linear, that is equivalent to see if you can estimate when the initial data is 0. Since your solution, which is known to exist, is continuous, you can apply Gronwall: en.wikipedia.org/wiki/… –  Willie Wong May 24 '13 at 7:13
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