I have been trying for some time now to understand the double $\delta$-function potential, as solved here. One question in particular is why we don't consider scattering "back" from the second potential.
Thus: assume we have two $\delta$-function potentials, and a particle coming from the left, arriving at the first potential. Then the particle is either reflected or transmitted, with some probability. But surely the same holds true at the other $\delta$-potential? I.e. the particle could meet the second potential, and get reflected back towards the first potential? And at the first potential it might again get either reflected (going back towards the second potential) or transmitted (going out from the first potential, adding to the first transmission).
And surely this process could continue indefintely (albeit with smaller and smaller probabilities). Should this not lead to a steady state problem, where one needs to take into account the probability that the particle "bounces around" a certain number of times before being either transmitted or reflected? If so, then why do we define the transmission/reflection coefficients simply as the ratio of the amplitudes of the initial reflection/transmission? Or is this possibility somehow contained in this coefficients?
I have looked around and found e.g.this Stack Exchange question, which mentions the $S$-matrix for a single $\delta$-function potential. Could such an approach be used here?