D. Damanik: Quantum Mechanics and Quasicrystals — Conjecture

Definitions and mathematical statement

For the real compact spectral-set setting intended by the source, a Cantorval is described by two conditions on $K\subset{\mathbb{R}}$: \[ K=\overline{\operatorname{int}_{{\mathbb{R}}}K}, \qquad\text{no connected component of }K\text{ is isolated}. \] Here an isolated component can be read as a component that is open in the relative topology of $K$, separated from all other components; the source does not further specify this convention. The closure and interior are in ${\mathbb{R}}$.

The conjecture asks whether there exist parameters $\lambda_1,\lambda_2$ for the source's operator $H$ such that \[ \sigma\left(H_{\lambda_1,\lambda_2}\right) \text{ is a Cantorval in the stated sense}. \] Neither $H$, its Hilbert space and domain, the one-dimensional sequences $s^\lambda$, nor the admissible parameter set is defined in the supplied statement. The research summary supplies only the separable-potential description \[ V(m,n)=s^{\lambda_1}(m)+s^{\lambda_2}(n) \] and calls the operator two dimensional. That does not determine its kinetic term, boundary conditions, or sampling sequence. A particular lattice Hamiltonian or Fibonacci-type sequence is not inferred.

The same summary says the spectrum is related to arithmetic sums of spectra of one-dimensional operators. Without the missing operator definition, the precise sum identity and its hypotheses cannot be supplied as part of the statement. The target remains existence of a Cantorval for this specific source operator, not merely existence of an abstract arithmetic sum with Cantorval topology.

Short English statement

Find parameter values for which the specified separable two-dimensional quasicrystal operator has a spectrum equal to the closure of its interior and with no isolated connected component. The operator and parameter domain are missing, so a complete standalone formulation needs source recovery.

The supplied summary reports no explicit example, proof, or disproof for this operator and labels the conjecture open. It cites general Cantorval literature as motivation but not a solution, and its final comparison is truncated. The OPEN-TRIAGE label and the 2016 source are retained with current status marked for review.

Sources

• Local UnsolvedMath registry, permanent ID 6600004, source code catalogue entry, "D. Damanik: Quantum Mechanics and Quasicrystals --- Conjecture," proposed year 2016; proposer metadata is blank. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals (2016), D. Damanik, "Quantum Mechanics and Quasicrystals," Conjecture 4 in source order, https://arxiv.org/abs/1604.06280; source-TeX extraction, accessed 2026-07-29.
• Mendes--Oliveira, Nonlinearity 7 (1994), as cited by the source for Cantorvals; no complete title or page range is supplied.
• The supplied summary mentions arXiv:2401.05372 and arXiv:2309.01589 for abstract Cantorval/achievable-set context; its comparison is truncated and supplies no operator-specific resolution.
• Source-specific provenance: the source labels the item a conjecture, current status NEEDS_REVIEW; arXiv source TeX and a CC BY journal version are available, while the arXiv item-specific reuse licence NEEDS_REVIEW. This qualification is separate from dataset attribution.