I am reading this paper and I would like to understand how the authors obtain the equations for the prefactors $C_\gamma^\pm$ (Eq. 14 of the paper).
They have to solve a diffusion problem in each region and they decompose the probability density $P_\gamma$ in two parts in order to find the solution, one depends on the coordinates of the center of mass of the system $P_X$ and the other one depends on the relative displacement $P_s$.
They consider a general function for $P_s$ in terms of an integral of complex exponentials. I don't understand a few things about this:
Why don't they consider a term like $C_{21}^- e^{-iQs}$ (Eq. 12) for $s>b/2$? I don't see any reason to neglect this term and not any of the others. They can reduce their system to a system of 4 equations with 4 unknows thanks to this fact and to taking $C_{12}^+=1$. How could one proceed in the general case? How could one get the missing two equations for the $C_\gamma$? (Besides matching of the derivatives at the boundary and the detailed balance conditions)
Intuitively I would consider conditions related to the vanishing of $P_\gamma$ at infinity but I don't see how this would be possible with this decomposition of $P_\gamma$ as an integral of complex exponentials.