Topics
Superlattices, Integral equations, Partial differential equations, Position-dependent mass, Scattering theory, Scattering matrix, Levinson's theorem, One dimensional Schrodinger equation

In a recent paper, Giust et al.1 showed how the elements of the 2×2 scattering matrix of a one-dimensional system composed of two subsystems can be expressed in terms of the scattering amplitudes of the individual subsystems to generalize Einstein’s composition law of velocities.

We note that this factorization property of the one-dimensional S-matrix (Eq. (21) in Ref. 1) was first derived by Aktosun2 for the one-dimensional Schrödinger equation and for the wave equation in nonhomogeneous, nondispersive mediums when the wave speed has the same asymptotic behavior at both ends of the real line. Aktosun presented this result in a very compact form using Λ-matrices that can be associated with the S-matrices. We write the on-shell S-matrix as

S=(TRLT),
(1)

with T as the transmission amplitude and L and R as the reflection...

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