Solving scattering problems in the half-line using methods developed for scattering in the full line

We reduce the solution of the scattering problem defined on the half-line $[0,\infty )$ by a real or complex potential $v\left(x\right)$ and a general homogeneous boundary condition at $x=0$ to that of the extension of $v\left(x\right)$ to the full line that vanishes for $x<0$. We find an explicit expression for the reflection amplitude of the former problem in terms of the reflection and transmission amplitudes of the latter. We obtain a set of conditions on these amplitudes under which the potential in the half-line develops bound states, spectral singularities, and time-reversed spectral singularities where the potential acts as a perfect absorber. We examine the application of these results in the study of the scattering properties of a $\delta$-function potential and a finite barrier potential defined in $[0,\infty )$, discuss optical systems modeled by these potentials, and explore the configurations in which these systems act as a laser or perfect absorber. In particular, we derive an explicit formula for the laser threshold condition for a slab laser with a single mirror and establish the surprising fact that a nearly perfect mirror gives rise to a lower threshold gain than a perfect mirror. We also offer a nonlinear extension of our approach which allows for utilizing a recently developed nonlinear transfer matrix method in the full line to deal with finite-range nonlinear scattering interactions defined in the half-line.