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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-4138-001.