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We reduce the solution of the scattering problem defined on the half-line$[0,\infty)$ by a real or complex potential $v(x)$ and a general homogenousboundary condition at $x=0$ to that of the extension of $v(x)$ to the full linethat vanishes for $x<0$. We find an explicit expression for the reflectionamplitude of the former problem in terms of the reflection and transmissionamplitudes of the latter. We obtain a set of conditions on these amplitudesunder which the potential in the half-line develops bound states, spectralsingularities, and time-reversed spectral singularities where the potentialacts as a perfect absorber. We examine the application of these results in thestudy of the scattering properties of a $\delta$-function potential and afinite barrier potential defined in $[0,\infty)$, discuss optical systemsmodeled by these potentials, and explore the configurations in which thesesystems act as a laser or perfect absorber. In particular, we derive anexplicit formula for the laser threshold condition for a slab laser with asingle mirror and establish the surprising fact that a nearly perfect mirrorgives rise to a lower threshold gain than a perfect mirror. We also offer anonlinear extension of our approach which allows for utilizing a recentlydeveloped nonlinear transfer matrix method in the full line to deal withfinite-range nonlinear scattering interactions defined in the half-line.

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