I am trying to figure out the scattering wave function for the following potential:
$$V(x,x')=-A \phi(x)\phi^*(x')$$ Such that the SE can be written as $$[\frac{\hbar^2\partial^2_x}{2m}-E]\psi = A\phi(x)\int dx'\phi^*(x')\psi(x')$$ This has a solution $$\psi(x)=\alpha e^{ikx}+\beta e^{-ikx}+\lambda[\int dx' K(x,x';E)\phi(x')\int dx''\phi(x'')\psi(x'')]$$ Where $K$ is the propagator as defined in Sakurai: $$K(x,x';E)=\frac{2m}{\hbar\sqrt{2mE}}e^{i|x-x'|\sqrt{2mE}/\hbar}$$ Back to the question, I am lost with. Based on this information how can I find a $\psi$ that satisfies the boundry conditions:
$$\psi(x\rightarrow-\infty)=e^{ikx}+re^{-ikx}$$ $$\psi(x\rightarrow\infty)=te^{ikx}$$
Not completely sure how to solve this. Supposedly, it can be assumed that $\phi$ goes to 0 as $x$ goes to $\infty$, which immediately implies the boundary conditions, but that does not seem clear to me why that happens