{
  "schema_version": 1,
  "problem_number": "OWR-4138-001",
  "title": "A Proof of the Nichols–Stolz Conjecture on the Almost-Sure Spectrum of the Bernoulli Displacement Model",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "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.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.SP",
    "math-ph"
  ],
  "keywords": [
    "random Schrödinger operators",
    "random displacement model",
    "Bernoulli displacement model",
    "almost-sure spectrum",
    "spectral gaps",
    "Schur complement",
    "Nichols–Stolz conjecture",
    "Oberwolfach Reports",
    "OWR-4138-001",
    "math.SP",
    "math-ph",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-09-29",
  "publication_date": "2026-09-29",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-29",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-4138-001/",
  "pdf_url": "https://eulersolve.org/papers/owr-4138-001/paper.pdf?v=f8ef386e1153",
  "doi": "10.5281/zenodo.23041940",
  "zenodo_record_url": "https://zenodo.org/records/23041940",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Proves Conjecture 6.2 of Nichols and Stolz for every λ ≠ 0, building on their reduction; Conjecture 6.3 (thin tails of the integrated density of states) is not addressed. A 2022 paper of Chulaevsky on localization for displacement models was not accessible to us. Unrefereed.",
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    "source.zip": {
      "sha256": "41672acd24c1c04e2158bf46c7789046a2c60cfec4397e5ac8c43b9252ee6325"
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    "verification_report.md": {
      "sha256": "6656e605a66eb3f6240133db3765793cbec4f152ac635551df9b4d34482d8de8"
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}