Multiplicity bound of singular spectrum for higher rank Anderson models

Abstract

In this work, we prove a bound on the multiplicity of the singular spectrum for a certain class of Anderson Hamiltonians. The operator in consideration is the form Hω=Δ+∑n∈ZdωnPn on the Hilbert space ℓ2(Zd), where Δ is discrete laplacian, Pn are projection onto ℓ2({x∈Zd:nili<xi≤(ni+1)li}) for some l1,⋯,ld∈N and {ωn}n are i.i.d. real bounded random variables following an absolutely continuous distribution. We prove that the multiplicity of the singular spectrum is bounded above by 2d−d independent of {li}i=1d. When li+1∉2N∪3N for all i and gcd(li+1,lj+1)=1 for i≠j, the singular spectrum is also simple.