On the unitarity of S-matrix in 1-d case

TL;DR

This paper demonstrates that the scattering S-matrix remains unitary in one-dimensional cases even when the potential approaches different limits at infinity, challenging previous claims that such conditions break unitarity.

Contribution

It clarifies the conditions under which the S-matrix remains unitary, correcting misconceptions caused by wave function normalization issues.

Findings

S-matrix is unitary despite different potential limits at infinity

Incorrect normalization can lead to false conclusions about unitarity

Provides clarification on scattering theory in 1D systems

Abstract

It is shown that the scattering S-matrix is unitary even if the scattering potential U(x) tends to different limits at plus and minus infinity. This result is in contrast to the statements of some authors which argue that the different potential values at infinity can break unitarity. The mistake may result from a wrong normalization of the wave functions. This work may be considered as a comment to some of those works.