bibkey: lindbladguerrero2025anderson authors: O. Lindblad and E. Guerrero year: 2025 title: Simple Eigenvalues and Non-vanishing Eigenvectors of the Anderson Model doi: null url: https://arxiv.org/abs/2512.00278v1 claim: “Conversely, we also ask whether every bad potential shares a nontrivial symmetry with the laplacian.” strata_touched:
- D5/S3/Combinatorics/Graph/AndersonSymmetryConverseRefutation license: citation-only triage: anchor
Verified locator
Source: https://arxiv.org/abs/2512.00278v1
The version-one PDF, pages 2, 3, 7 and 8, supplies the definitions, question and symmetry lemma quoted below.
The model and badness
Section 1, page 2, writes
[ (H_t\psi)(j)=\sum_{k\sim j}(\psi(j)-\psi(k))+t\omega_j\psi(j). ]
Definition 1.1, page 2:
We say that V is a good potential if H_t = Δ + tV has simple eigenvalues and non-vanishing eigenvectors for all but finitely many t values. We say that V is a bad potential if H_t fails to satisfy at least one of these conditions for any t ∈ ℝ.
Section 1, page 2, describes
To explain the discrepancy, in section 3 we show shared symmetries — orthogonal matrices that commute with both Δ and V — lead to the failure of our conditions.
The converse question
After Theorem 1.3, page 3:
Conversely, we also ask whether every bad potential shares a nontrivial symmetry with the laplacian.
Section 4, page 8:
In general, we do not know whether the converse of lemma 3.2 is true.
Lemma 3.2, page 7, assumes an orthogonal O ≠ I commuting with the two real symmetric matrices. Its first branch assumes O² ≠ I; its second assumes Oe_j = e_j for some vertex. Each branch excludes O = −I. Accordingly, the converse question is encoded with O ≠ I and O ≠ −I. Merely allowing −I would make the conclusion hold for every potential.
On the cycle, Mathlib’s cycleGraph and lapMatrix give the displayed sum of neighbour differences. Simple spectrum is expressed by algebraic multiplicity one for each complex characteristic-polynomial root; non-vanishing requires every complex nonzero eigenvector to be nonzero at every vertex. The normalized values {−1,1} represent any two distinct potential values: V ↦ αV + βI, α ≠ 0, reparametrizes t by αt and shifts the eigenvalues by βt while preserving the eigenvectors and shared commutants.
The eight-cycle potential (1,1,1,1,−1,1,−1,−1) has a persistent nodal eigenvector branch and scalar shared commutant. These are conclusions of the accompanying refutation, rather than results attributed to the paper.