Absolutely Continuous Spectrum for Random Schroedinger Operators on the Bethe Strip

TL;DR

This paper proves that for small disorder, Anderson models on the Bethe strip exhibit absolutely continuous spectrum, indicating extended states, with the spectrum's density of states being smooth within certain energy intervals.

Contribution

It establishes the presence of absolutely continuous spectrum for Anderson models on the Bethe strip at small disorder levels, extending understanding of spectral properties in such structures.

Findings

Absolutely continuous spectrum exists for small disorder.

Spectrum's density of states is continuously differentiable.

Results hold for a broad class of energy intervals.

Abstract

The Bethe Strip of width $m$ is the cartesian product $\B\times\{1,...,m\}$, where $\B$ is the Bethe lattice (Cayley tree). We prove that Anderson models on the Bethe strip have "extended states" for small disorder. More precisely, we consider Anderson-like Hamiltonians $\;H_\lambda=\frac12 \Delta \otimes 1 + 1 \otimes A + \lambda \Vv$ on a Bethe strip with connectivity $K \geq 2$, where $A$ is an $m\times m$ symmetric matrix, $\Vv$ is a random matrix potential, and $\lambda$ is the disorder parameter. Given any closed interval $I\subset (-\sqrt{K}+a_{\mathrm{max}},\sqrt{K}+a_{\mathrm{min}})$, where $a_{\mathrm{min}}$ and $a_{\mathrm{max}}$ are the smallest and largest eigenvalues of the matrix $A$, we prove that for $\lambda$ small the random Schr\"odinger operator $\;H_\lambda$ has purely absolutely continuous spectrum in $I$ with probability one and its integrated density of states is continuously differentiable on the interval $I$.