Useful equation of tridiagonal matrices in application to electron transport through a quantum wire

TL;DR

This paper derives an analytical formula linking cofactors and determinants of symmetric tridiagonal matrices, facilitating calculations of electron transmittance in quantum wires and demonstrating the equivalence of two computational methods.

Contribution

It introduces a new nonlinear formula for symmetric tridiagonal matrices, bridging the evolution operator and Green's function methods in quantum transport analysis.

Findings

Derived an analytical nonlinear formula for symmetric tridiagonal matrices

Established the equivalence between evolution operator and Green's function methods

Applied the formula to calculate transmittance in quantum wires

Abstract

In this paper the transmittance through a quantum wire connected with two electron reservoirs is calculated and non-trivial transformation between the evolution operator method and the Green's function technique is reported. To show this equivalence an analytical nonlinear formula which concerns symmetrical tridiagonal matrices is proofed. This formula connects the cofactor and three determinants of tridiagonal matrices.