On the electron scattering on the one-dimensional complexes: the vertex amplitudes method

TL;DR

This paper introduces a new theoretical method for analyzing electron scattering on one-dimensional quantum graphs, simplifying the problem to solving linear equations for vertex amplitudes and providing detailed insights into transmission resonances.

Contribution

A novel vertex amplitudes method that reduces the electron transport problem on quantum graphs to linear algebra, enabling comprehensive analysis of scattering properties.

Findings

Transmission and reflection amplitudes are expressed via a single matrix.

The approach simplifies the calculation of wave functions and probability currents.

Detailed analysis of transmission resonances compared with existing results.

Abstract

The problem of electron scattering on the one-dimensional complexes is considered. We propose a novel theoretical approach to solution of the transport problem for a quantum graph. In the frame of the developed approach the solution of the transport problem is equivalent to the solution of a linear system of equations for the \emph{vertex amplitudes} $\mathbf{\Psi}$. All major properties, such as transmission and reflection amplitudes, wave function on the graph, probability current, are expressed in terms of one $\mathbf{\Gamma}$-matrix that determines the transport through the graph. The transmission resonances are analyzed in detail and comparative analysis with known results is carried out.