Schrödinger operators with decaying randomness - Pure point spectrum

2018/08/17 by Mallick, Anish, Dolai, Dhriti Ranjan
#35J10 #47B80 #81Q10 #82B44 #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1808.05822

Abstract

Here we show that for Schrödinger operator with decaying random potential with fat tail single site distribution, the negative spectrum shows a transition from essential spectrum to discrete spectrum. We study the Schrödinger operator H^ω=-Δ+∑_n∈ℤ^da_nω_nχ__(0,1]^d(x-n) on L²(ℝ^d). Here we take a_n=O(|n|^-α) for large n where α>0, and \ω_n\_n∈ℤ^d are i.i.d real random variables with absolutely continuous distribution μ such that (dμ)/(dx)(x)=O(|x|^-(1+δ))~as~|x|→∞, for some δ>0. We show that H^ω exhibits exponential localization on negative part of spectrum independent of the parameters chosen. For αδ≤ d we show that the spectrum is entire real line almost surely, but for αδ>d we have σ_ess(H^ω)=[0,∞) and negative part of the spectrum is discrete almost surely. In some cases we show the existence of the absolutely continuous spectrum.

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