A Proof of the Nichols–Stolz Conjecture on the Almost-Sure Spectrum of the Bernoulli Displacement Model

The Bernoulli displacement model is the random Schrödinger operator h_{ω,λ} = h_0 + V_ω on ℓ²(Z) in which each cell of two neighbouring sites carries one single-site potential λ ≠ 0. The potential sits on the left or on the right site of the cell according to an independent Bernoulli variable ω_k. Nichols and Stolz determined the almost-sure spectrum Σ_λ for 0 < |λ| ≤ 2. They conjectured (Oberwolfach Report 55/2009; J. Spectral Theory 1 (2011), Conjecture 6.2) that Σ_λ = σ(h_{ω*,λ}) ∪ σ(h_{ω¹,λ}) for every λ ≠ 0, where the configurations ω* and ω¹ give potentials of period four and two. For |λ| > 2 this means that Σ_λ consists of exactly six bands. We prove the conjecture. In fact, the inclusion σ(h_{ω,λ}) ⊂ σ(h_{ω*,λ}) ∪ σ(h_{ω¹,λ}) = {E ∈ R : |E(E − λ) − 2| ∈ [0, 2] ∪ [|λ|, √(λ² + 4)]} holds for every configuration ω, not only almost surely. The proof uses the reduction of Nichols and Stolz, under which h_{ω,λ}(h_{ω,λ} − λ) − 2 becomes a direct sum of two discrete Schrödinger operators with potentials ±s, where s_k = λ(ω_k − ω_{k+1}). It combines this reduction with an elementary Schur-complement argument. The argument works because s never takes the same nonzero value at two neighbouring sites. This is an unrefereed note.