Some linear differential expressions for an electron scattering problem in a field of the one-dimentional arbitrary potential

TL;DR

This paper derives a linear differential system to analyze electron scattering in one-dimensional arbitrary potentials, reducing the problem to a classical Cauchy problem and providing an explicit functional for transmission amplitude dependence.

Contribution

It introduces a new linear differential framework for electron scattering problems in arbitrary potentials, linking it to the Cauchy problem for Schrödinger's equation.

Findings

Reduced scattering problem to Cauchy problem

Derived explicit functional for transmission amplitude

Applicable to arbitrary one-dimensional potentials

Abstract

The linear system of differential equations for determination of transmission and reflection amplytudes of scattered electron in the field of one dimensional arbitrary potential is obtained. It is shown that in general the scattering problem can be reduced to the Caushy problem for the stationary Shrodinger equation. An explicit functional for a dependance of transmission amplitude from scattered field is found.