Periodic structure and Schrodinger operators for codings of circle rotations
Abstract
We consider, for any irrational α and interval I ⊂ 𝕋, the 2-interval coding subshift X(I, α) induced by coding orbits under repeated rotation by α via membership in I or Ic. Each sequence c ∈ X(I, α) has an associated Schrödinger operator Hc, and in \cite{kaminaga} it was proved that if the continued fraction of α has digits with limsup at least $4$, then almost every c ∈ X(I, α) has so-called $3$-block Gordon structure, which implies that the operator Hc has no eigenvalues. We significantly improve this result by completely characterizing almost-sure $3$-block Gordon structure, proving that in fact it holds for all (α,I) except for a countable set of pairs $(α, |I|)$ where α is Möbius equivalent to the silver mean and |I| ∈ ℤα+ f(α) where f(α) is a specific infinite series taking value either 1/2, α/2, or (α+1)/2. We also show that for a set of α of full measure, and for every I, the set of points whose orbit codings do not have $3$-block Gordon structure has Hausdorff dimension bounded away from $1$.
Related
- Band edges of periodic Schrödinger operators are generically isolated and nondegenerate
- Irreducibility of the Bloch variety for periodic Schrödinger operators in arbitrary dimension
- Polynomial Spectral Gap Estimates for Finitely Differentiable Quasi-Periodic Schrödinger Operators
- Two-scale estimates for limit-periodic Schrödinger operators
- Convergence Rate of Birkhoff Average for Toral Quasi-Periodic Rotations and Applications
BibTeX
@misc{hetzel2026periodic,
title = {{Periodic structure and Schrodinger operators for codings of circle rotations}},
author = {Luke Hetzel and Ronnie Pavlov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00301},
eprint = {2609.00301},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00301},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Periodic structure and Schrodinger operators for codings of circle rotations
AU - Luke Hetzel
AU - Ronnie Pavlov
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00301
UR - https://arxiv.org/abs/2609.00301
AB - We consider, for any irrational α and interval I ⊂ 𝕋, the 2-interval coding subshift X^(I, α) induced by coding orbits under repeated rotation by α via membership in I or I^c. Each sequence c ∈ X^(I, α) has an associated Schrödinger operator H_c, and in \cite{kaminaga} it was proved that if the continued fraction of α has digits with limsup at least $4$, then almost every c ∈ X^(I, α) has so-called $3$-block Gordon structure, which implies that the operator H_c has no eigenvalues. We significantly improve this result by completely characterizing almost-sure $3$-block Gordon structure, proving that in fact it holds for all (α,I) except for a countable set of pairs $(α, |I|)$ where α is Möbius equivalent to the silver mean and |I| ∈ ℤα+ f(α) where f(α) is a specific infinite series taking value either 1/2, α/2, or (α+1)/2. We also show that for a set of α of full measure, and for every I, the set of points whose orbit codings do not have $3$-block Gordon structure has Hausdorff dimension bounded away from $1$.
ER -