Absolutely Continuous Spectrum for Random Schroedinger Operators on the Bethe Strip

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 $\;H_λ=\frac12 Δ\otimes 1 + 1 \otimes A + λ\Vv$ on a Bethe strip with connectivity $K\ge 2$, 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 $I\subset (-\sqrt{K}+a_{\max },\sqrt{K}+a_{\min })$, where $a_{\min }$ and $a_{\max }$ are the smallest and largest eigenvalues of the matrix $A$, we prove that for $\lambda$ small the random Schrödinger 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 $I$.

References in corpus (1)

• Ballistic Behavior for Random Schrödinger Operators on the Bethe Strip

Cited by in corpus (13)

• Anderson transition at 2 dimensional growth rate on antitrees and spectral theory for operators with one propagating channel
• Absolutely continuous spectrum for multi-type Galton Watson trees
• Absolutely Continuous Spectrum for Random Schroedinger Operators on Tree-Strips of finite cone type
• Absolutely Continuous Spectrum for random Schroedinger operators on the Fibonacci and similar tree-strips
• Spectral Theory of one-channel operators and application to absolutely continuous spectrum for Anderson type models
• Power law logarithmic bounds of moments for long range operators in arbitrary dimension
• Delocalization and continuous spectrum for ultrametric random operators
• Resonant delocalization on the Bethe strip
• A central limit theorem for products of random matrices and GOE statistics for the Anderson model on long boxes
• On quantum percolation in finite regular graphs
• A supersymmetric hierarchical model for weakly disordered $3d$ semimetals
• Jakšić-Last Theorem for Higher Rank Perturbations
• On absolutely continuous spectrum for one-channel unitary operators