Localization for one-dimensional random potentials with large local fluctuations

TL;DR

This paper investigates how large local fluctuations in one-dimensional random potentials affect wave function localization, revealing new energy dependencies and superlocalization phenomena.

Contribution

It introduces new localization length behaviors in 1D Schrödinger equations with highly fluctuating random potentials, including superlocalization effects.

Findings

Localization length scales as E/ln E in some regimes

Localization length scales as E^{μ/2} for 0<μ<2

Superlocalization with faster-than-exponential decay

Abstract

We study the localization of wave functions for one-dimensional Schr\"odinger Hamiltonians with random potentials $V(x)$ with short range correlations and large local fluctuations such that $\int\D{x} \smean{V(x)V(0)}=\infty$. A random supersymmetric Hamiltonian is also considered. Depending on how large the fluctuations of $V(x)$ are, we find either new energy dependences of the localization length, $\ell_\mathrm{loc}\propto{}E/\ln{E}$, $\ell_\mathrm{loc}\propto{}E^{\mu/2}$ with $0<\mu<2$ or $\ell_\mathrm{loc}\propto\ln^{\mu-1}E$ for $\mu>1$, or superlocalization (decay of the wave functions faster than a simple exponential).