Spectral Dependence of Degree of Localization of Eigenfunctions of the 1D Schrodinger Equation with a Peacewise-Constant Random Potential

TL;DR

This paper develops a perturbation theory to analyze how the localization of eigenfunctions in a 1D Schrödinger equation with random potential depends on spectral properties, providing analytical and computational insights.

Contribution

It introduces a novel perturbation approach for joint Green's functions and derives analytical expressions for localization characteristics in a 1D disordered system.

Findings

Analytical expressions for spectral dependence of localization

Confirmation of theoretical results through computer experiments

Definition of localization length in the context of random potentials

Abstract

The perturbation theory is developed for joint statistics of the advanced and retarded Green's functions of the 1D Schrodinger equation with a piecewise-constant random potential. Using this method, analytical expressions are obtained for spectral dependence of the degree of localization and for the limiting (at $t\rightarrow\infty$) probability to find the particle at the point it was located at $t = 0$ (Andeson criterion). Definition of the localization length is introduced. The computer experiments confirming correctness of the calculations are described.