Spectra of Non-Self-Adjoint Almost Mathieu Matrices and the Scottish Flag Operator
Abstract
For N≥ 3 and a potential phase ϑ∈ℝ, we study the non-self-adjoint almost Mathieu matrix obtained by multiplying the discrete Laplacian by a complex phase with angle φ∈ℝ, AN(φ,ϑ)=eiφ(S+S−1)/2+diag(cos(2πj/N+ϑ))j∈ℤ/Nℤ, where S ej=ej+1 is the periodic shift on ℂN. We derive a Chambers formula and isolate the part QN,φ of the characteristic polynomial that depends only on N and φ, but not on ϑ or on a change of boundary conditions for the shift operator. We then show, for every N, that the zeros of QN,φ lie on the two perpendicular lines eiφ/2ℝ∪ ei(φ/2+π/2)ℝ. For even N, the same property holds for the matrices AN(φ,ϑ) with ϑ∈ 2πℤ/N, and we compute their limiting eigenvalue measure explicitly. For φ∈[−π,π], the eigenvalue distribution approximates elliptic-integral densities with masses 1−|φ|/π and |φ|/π, and maximal radii 2|cos(φ/2)| and 2|sin(φ/2)|, respectively. At φ=π/2, the central polynomial QN,φ factors into positive quartic factors. This proves that the Scottish flag matrix, after Trefethen and Chapman, has its spectrum on the two diagonal lines of the saltire.
Related
- A Mixed-Adjoint Framework for 2 × 2 Operator Matrices and Their Numerical Radii
- Cantor Spectrum via a Reducibility-Duality Bridge for the Mosaic Almost Mathieu Operator
- The Spectrum of the Discrete Mathieu Operator with Nonreal Coupling
- Persistence of Unitary Almost Mathieu Operator Gaps Ten Dry Martinis, Five on the Rim
- Adjoint Transfer and Order-Lifting Dichotomies for 2 x 2 Matrices over Prime-Power Rings
BibTeX
@misc{becker2026spectra,
title = {{Spectra of Non-Self-Adjoint Almost Mathieu Matrices and the Scottish Flag Operator}},
author = {Simon Becker and Izak Oltman},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30191},
eprint = {2608.30191},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30191},
note = {CC0},
}
RIS
TY - UNPB
TI - Spectra of Non-Self-Adjoint Almost Mathieu Matrices and the Scottish Flag Operator
AU - Simon Becker
AU - Izak Oltman
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30191
UR - https://arxiv.org/abs/2608.30191
AB - For N≥ 3 and a potential phase ϑ∈ℝ, we study the non-self-adjoint almost Mathieu matrix obtained by multiplying the discrete Laplacian by a complex phase with angle φ∈ℝ, A_N(φ,ϑ)=e^iφ(S+S^−1)/2+diag(cos(2πj/N+ϑ))_j∈ℤ/Nℤ, where S e_j=e_j+1 is the periodic shift on ℂ^N. We derive a Chambers formula and isolate the part Q_N,φ of the characteristic polynomial that depends only on N and φ, but not on ϑ or on a change of boundary conditions for the shift operator. We then show, for every N, that the zeros of Q_N,φ lie on the two perpendicular lines e^iφ/2ℝ∪ e^i(φ/2+π/2)ℝ. For even N, the same property holds for the matrices A_N(φ,ϑ) with ϑ∈ 2πℤ/N, and we compute their limiting eigenvalue measure explicitly. For φ∈[−π,π], the eigenvalue distribution approximates elliptic-integral densities with masses 1−|φ|/π and |φ|/π, and maximal radii 2|cos(φ/2)| and 2|sin(φ/2)|, respectively. At φ=π/2, the central polynomial Q_N,φ factors into positive quartic factors. This proves that the Scottish flag matrix, after Trefethen and Chapman, has its spectrum on the two diagonal lines of the saltire.
ER -