The spatial statistical properties of wave functions in a disordered finite one-dimensional sample

TL;DR

This paper analytically derives the Laplace transform of the distribution of inverse spatial size of wave functions in a finite 1D disordered system, revealing their spatial statistical properties.

Contribution

It provides an analytical calculation of the Laplace transform of the distribution of wave function spatial sizes in a finite disordered 1D sample, a novel insight into their statistical behavior.

Findings

Analytical expression for the Laplace transform of $P()$

Insights into the spatial distribution of wave functions in disordered systems

Enhanced understanding of wave function localization in finite samples

Abstract

For a given wave function one can define a quantity $\mu_E$ having a meaning of its inverse spatial size. The Laplace transform of the distribution function $P(\mu_E)$ is calculated analytically for a 1D disordered sample with a finite length $L$.