D. Damanik: Quantum Mechanics and Quasicrystals — Conjecture

Definitions and mathematical statement

For $x\in\{0,1\}^{{\mathbb{Z}}}$ and an $n$-element set $\tau=\{t_1<\cdots<t_n\}\subset{\mathbb{Z}}$, let \[ p_x(\tau)=\#\{(x_{m+t_1},\ldots,x_{m+t_n}):m\in{\mathbb{Z}}\},\qquad p_x^*(n)=\sup_{|\tau|=n}p_x(\tau). \] Under the standard complexity convention, pattern-Sturmian means $p_x^*(n)=2n$ for every integer $n\geq1$. This is maximal pattern complexity, not ordinary consecutive-block complexity $n+1$.

For a real nonzero coupling $\lambda$, define the bounded self-adjoint operator on $\ell^2({\mathbb{Z}})$ by \[ (H\psi)(m)=\psi(m+1)+\psi(m-1)+\lambda x_m\psi(m). \] The proposed spectral conclusion is that $\sigma(H)$ is a Cantor set of zero one-dimensional Lebesgue measure and that every spectral measure is singular continuous. Here a Cantor set means a nonempty compact perfect nowhere-dense subset of ${\mathbb{R}}$. For $\psi\in\ell^2({\mathbb{Z}})$ its spectral measure is $\mu_\psi(B)=\langle\psi,E_H(B)\psi\rangle$, where $E_H$ is the spectral projection measure; singular continuous means singular with respect to Lebesgue measure and without atoms.

The intended quantification is over the specified pattern-Sturmian potentials and their coupling parameters. The source does not state any domain or exclusion for $\lambda$: self-adjoint spectral measures require real $\lambda$, and the proposed conclusion cannot include $\lambda=0$, which gives the free discrete operator. These restrictions are therefore explicit necessary clarifications, not silently supplied source hypotheses.

The research summary refers to all recurrent pattern-Sturmian potentials, whereas the statement does not require recurrence. In the usual two-sided convention, recurrence means every finite block appearing in $x$ reappears arbitrarily far in both directions. Whether this hypothesis is part of the source's pattern-Sturmian definition or an additional restriction must be checked. No proof for all sequences is inferred from results for Toeplitz or circle-map subclasses.

Short English statement

Do pattern-Sturmian binary Schrödinger potentials at nonzero real coupling always have zero-measure Cantor spectrum and purely singular-continuous spectral measures? The coupling and recurrence hypotheses are missing or inconsistent across the supplied fields.

Sources

• Local UnsolvedMath registry, permanent ID 6600002, exact source code catalogue entry, "D. Damanik: Quantum Mechanics and Quasicrystals---Conjecture"; proposed year 2016, no proposer in metadata. Registry status partially solved, classification PARTIAL-PROGRESS.
• Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016), conjecture 2 in source order, D. Damanik section, https://arxiv.org/abs/1604.06280. Supplied source-TeX extraction accessed July 29, 2026.
• The supplied research summary reports the Toeplitz family case proved in arXiv:1511.03834, without a title or author list. It cites a structural classification by Le--Pavlov--Schlortt (2025), related source code AMR-065-0001, and mentions circle-map rotation codings and nearly simple Toeplitz classes. It expressly says full generality for recurrent pattern-Sturmian potentials is not yet proved. The claimed classification and subclass coverage are not independently verified here.
• Provenance flags current status NEEDS_REVIEW. Its rights note says source TeX and a CC BY journal version are available, while the arXiv item-specific reuse license remains NEEDS_REVIEW.