The MS Unidirectional Current as a Generalization of the ABC Absorption Current

Abstract

For a free scalar particle in one spatial dimension, in a single-pass half-line geometry, we compare the absorbing Robin boundary condition with the generalized Marchewka--Schuss (MS) unidirectional-current family. Robin absorption is characterized by a single parameter $\kappa>0$: once $\kappa$ is specified, its momentum-dependent absorption profile is fixed. In the generalized MS construction, by contrast, the detector response is described by a spectral function $\lambda(k)$, subject to the positivity and subnormalization conditions of the detection law. We show that every Robin choice of $\kappa$ corresponds to the particular MS calibration $\lambda_\kappa(k)=\pi\kappa/(k+\kappa)^2$, which reproduces the complete Robin absorption current for every admissible one-sided spectral amplitude and at every time. Thus the entire one-parameter Robin absorption family is contained within the generalized MS family. The MS construction is more general: the choice $\lambda_{\rm full}(k)=\pi/(4k)$ gives unit absorption efficiency for every wave number, which cannot be achieved by any fixed Robin parameter. In this precise sense, the generalized MS detector process extends the Robin absorbing-boundary family. We also compare their asymptotic behavior in the ballistic fixed-ray regime and in the fixed-distance late-time limit.