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The Dry Ten Martini Problem for $C^2$ cosine-type quasiperiodic Schrödinger operators

Lingrui Ge, Yiqian Wang, Jiahao Xu

TL;DR

This work resolves The Dry Ten Martini Problem for $C^2$ cosine-type quasiperiodic Schrödinger operators at large coupling and Diophantine frequencies, showing that all spectral gaps predicted by the Gap-Labelling Theorem are open, and further proves the spectrum is homogeneous and the IDS is absolutely continuous. The authors develop a robust four-step strategy that labels gaps via resonances of cosine-type critical points, proves label invariance in the coupling, links rotation numbers to gap labels, and derives precise large-$\lambda$ gap asymptotics. A pivotal contribution is the Induction Theorem for $C^2$ cosine-type potentials, which governs the evolution of the angle function and critical points across scales, enabling a new, geometry-driven gap labeling that aligns with Johnson–Moser labeling. The results yield quantitative gap control, establish the relationship between critical-point distances and the IDS, and demonstrate the global regularity consequences (homogeneity and absolute continuity) of the spectrum for these low-regularity models.

Abstract

This paper solves ``The Dry Ten Martini Problem'' for $C^2$ cosine-type quasiperiodic Schrödinger operators with large coupling constants and Diophantine frequencies, a model originally introduced by Sinai in 1987 \cite{sinai}. This shows that the analyticity assumption on the potential is not essential for obtaining a dry Cantor spectrum and can be replaced by a certain geometric condition in the low regularity case. In addition, we prove the homogeneity of the spectrum and the absolute continuity of the integrated density of states (IDS).

The Dry Ten Martini Problem for $C^2$ cosine-type quasiperiodic Schrödinger operators

TL;DR

This work resolves The Dry Ten Martini Problem for cosine-type quasiperiodic Schrödinger operators at large coupling and Diophantine frequencies, showing that all spectral gaps predicted by the Gap-Labelling Theorem are open, and further proves the spectrum is homogeneous and the IDS is absolutely continuous. The authors develop a robust four-step strategy that labels gaps via resonances of cosine-type critical points, proves label invariance in the coupling, links rotation numbers to gap labels, and derives precise large- gap asymptotics. A pivotal contribution is the Induction Theorem for cosine-type potentials, which governs the evolution of the angle function and critical points across scales, enabling a new, geometry-driven gap labeling that aligns with Johnson–Moser labeling. The results yield quantitative gap control, establish the relationship between critical-point distances and the IDS, and demonstrate the global regularity consequences (homogeneity and absolute continuity) of the spectrum for these low-regularity models.

Abstract

This paper solves ``The Dry Ten Martini Problem'' for cosine-type quasiperiodic Schrödinger operators with large coupling constants and Diophantine frequencies, a model originally introduced by Sinai in 1987 \cite{sinai}. This shows that the analyticity assumption on the potential is not essential for obtaining a dry Cantor spectrum and can be replaced by a certain geometric condition in the low regularity case. In addition, we prove the homogeneity of the spectrum and the absolute continuity of the integrated density of states (IDS).

Paper Structure

This paper contains 20 sections, 31 theorems, 314 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main Strategy
  3. Transformation Between SL(2,$\mathbb{R}$) Cocycles and Schrödinger Cocycles
  4. New labels of the spectral gaps
  5. Induction Theorem for $C^2$ Cosine Type
  6. Some remarks on the Induction Theorem
  7. A baby model
  8. The properties $c_{n,j}$ with respect to $t$
  9. The definition of the new labels of gaps
  10. Proof of Theorems \ref{['Th2']}, \ref{['Th4']}, and \ref{['Th5']}
  11. Large Coupling Asymptotics of the IDS
  12. Continuity of the Spectral Gaps
  13. Proof of Theorem \ref{['Th2']}
  14. Proof of Theorem \ref{['Th4']}
  15. Proof of Theorem \ref{['Th5']}
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\alpha \in DC$ and $v$ be a $C^2$ cosine-type potential. There exists a constant $\lambda_0(\alpha, v)$ such that if $\lambda \geq \lambda_0$, then The Dry Ten Martini Problem holds for $H_{\alpha, \lambda v, x}$ for any $x \in \mathbb{T}$.

Figures (5)

  • Figure 1: Graphs of $g_{n+1}$ in $I_{n+1,1}$: the jump part may move outside $I_{n+1,1}$ if $|d_{n}|$ increases as $t$ changes. The first and the third pictures correspond to Case 2 and 1, respectively.
  • Figure 2: The Case 2: $g_{n+1}\ {\rm\ (mod\ \pi)}$ may have two zeroes (the first and fifth pictures), one zero (the second and fourth pictures) or no zeroes (the third picture, corresponds to Case 3) in $x\in I_{n+1,1}$ if $d_{n}$ changes its sign as $t$ changes.
  • Figure 3: The graph of $g_{n+1}-g_{n}=\tilde{\phi}(x,t)$, the restriction of the complete graph of approximate arc tangent function in $n_1\alpha+I_{n+1,1}$, depends on the position of $c^{*}_n$, where $c^{*}_n$ is the middle point of the jump part. The Case that $g_{n+1}-g_{n} {\rm \ mod}\ \pi$ is small corresponds to $c^{*}_n\not\in n_1\alpha+I_{n+1,1}$; and the Case that $g_{n+1}-g_{n}$ is a pulse function corresponds to $c^{*}_n\in n_1\alpha+I_{n+1,1}$.
  • Figure 4: In the domain $I_{n,1}\bigcup T^{-k}I_{n,2},$ red dashed line is $\arctan[\|A_k(x,t)\|^2 \tan(g_{n,2}(x+k\alpha,t))]+\frac{\pi}{2}$, the yellow dashed line is $g_{n,1}(x,t)$; In the domain $I_{n,2}\bigcup T^{k}I_{n,1},$ yellow dashed line is $\arctan[\|A_k(x,t)\|^2 \tan(g_{n,1}(x-k\alpha,t))]+\frac{\pi}{2}$ and the red dashed line is $g_{n,2}(x,t).$ The orange part is yellow part $+$ red part, which represents $g_{n+1}.$
  • Figure 5: Graphs of $g_{n+1}$ as a function of $t$ in $I_{n+1}$:

Theorems & Definitions (54)

  • Theorem 1
  • Remark 1
  • Definition 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Proposition 1: z1
  • Remark 2
  • ...and 44 more