Band edge limit of the scattering matrix for quasi-one-dimensional discrete Schr\"odinger operators

TL;DR

This paper investigates the behavior of scattering matrices at band edges for quasi-one-dimensional discrete Schrödinger operators, providing explicit formulas and analyzing the limits of reflection coefficients.

Contribution

It introduces explicit expressions for the analytic continuation of transmission coefficients and calculates the limits of reflection coefficients at band edges.

Findings

Explicit formulas for extended transmission coefficients.

Calculated limits of reflection coefficients at band edges.

Analytic continuation of scattering data to spectral band edges.

Abstract

This paper is about the scattering theory for one-dimensional matrix Schr\"odinger operators with a matrix potential having a finite first moment. The transmission coefficients are analytically continued and extended to the band edges. An explicit expression is given for these extensions. The limits of the reflection coefficients at the band edges is also calculated.