Multiplicity bound of Singular Spectrum for higher rank Anderson models

Abstract

In this work, we prove a bound on multiplicity of the singular spectrum for certain class of Anderson Hamiltonians. The class of operator is $H^{\omega }=\Delta +{\sum }_{n\in {\mathbb{Z}}^d}{\omega }_n{P}_n$ on the Hilbert space ${\ell }^2({\mathbb{Z}}^d)$, where $\Delta$ is discrete laplacian, $P_n$ are projection onto ${\ell }^2(\{x\in {\mathbb{Z}}^d:n_i{l}_i<x_i\le (n_i+1)l_i\})$ for some $l_1,\cdots ,l_d\in \mathbb{N}$ and $\{{\omega }_n{\}}_n$ are i.i.d real bounded random variables following absolutely continuous distribution. We prove that the multiplicity of singular spectrum is bounded above by $2^d-d$ independent of $\{l_i{\}}_{i=1}^d$. When $l_i+1\notin 2\mathbb{N}\cup 3\mathbb{N}$ for all $i$ and $gcd(l_i+1,l_j+1)=1$ for $i\ne j$, we also prove that the singular spectrum is simple.

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References in corpus (7)

• New characterizations of the region of complete localization for random Schrödinger operators
• Simplicity of eigenvalues in the Anderson model
• Random Dirac operators with time-reversal symmetry
• Simplicity of eigenvalues in Anderson-type models
• Eigenvalue statistics for random Schrodinger operators with non rank one perturbations
• Jakšić-Last Theorem for Higher Rank Perturbations
• Boosted Simon-Wolff Spectral Criterion and Resonant Delocalization

Cited by in corpus (3)

• Multiplicity theorem of singular Spectrum for general Anderson type Hamiltonian
• On multiplicity of spectrum for Anderson type operators with higher rank perturbations
• Global multiplicity bounds and Spectral Statistics Random Operators