Global Bounds for the Lyapunov Exponent and the Integrated Density of States of Random Schr\"odinger Operators in One Dimension

TL;DR

This paper establishes bounds for the Lyapunov exponent and the integrated density of states of one-dimensional random Schr"odinger operators, demonstrating decay rates at high energies based on scattering data, with applications to the Kronig-Penney model.

Contribution

It provides new bounds for key spectral quantities of 1D random Schr"odinger operators using scattering data, applicable to models like Kronig-Penney.

Findings

Both $ ext{Lyapunov exponent}$ and $N(E)-rac{ ext{sqrt}(E)}{ ext{pi}}$ decay at least like $1/ ext{sqrt}(E)$ at high energies.

Bounds depend only on scattering data of the single-site potential.

Results apply to models with potentials of identical shape and random coupling constants.

Abstract

In this article we prove an upper bound for the Lyapunov exponent $\gamma(E)$ and a two-sided bound for the integrated density of states $N(E)$ at an arbitrary energy $E>0$ of random Schr\"odinger operators in one dimension. These Schr\"odinger operators are given by potentials of identical shape centered at every lattice site but with non-overlapping supports and with randomly varying coupling constants. Both types of bounds only involve scattering data for the single-site potential. They show in particular that both $\gamma(E)$ and $N(E)-\sqrt{E}/\pi$ decay at infinity at least like $1/\sqrt{E}$. As an example we consider the random Kronig-Penney model.