scholarly journals Anderson localization for the almost Mathieu operator in the exponential regime

2015 ◽  
Vol 5 (1) ◽  
pp. 89-112 ◽  
Author(s):  
Wencai Liu ◽  
Xiaoping Yuan
Keyword(s):  
Anderson Localization ◽  
Mathieu Operator ◽  
Almost Mathieu Operator

scholarly journals Almost Mathieu operators with completely resonant phases

2018 ◽  
Vol 40 (7) ◽  
pp. 1875-1893
Author(s):  
WENCAI LIU
Keyword(s):  
Anderson Localization ◽  
Continued Fraction ◽  
Mathieu Operator ◽  
Almost Mathieu Operator

Let $\unicode[STIX]{x1D6FC}\in \mathbb{R}\backslash \mathbb{Q}$ and $\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})=\limsup _{n\rightarrow \infty }(\ln q_{n+1})/q_{n}<\infty$, where $p_{n}/q_{n}$ is the continued fraction approximation to $\unicode[STIX]{x1D6FC}$. Let $(H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}u)(n)=u(n+1)+u(n-1)+2\unicode[STIX]{x1D706}\cos 2\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D703}+n\unicode[STIX]{x1D6FC})u(n)$ be the almost Mathieu operator on $\ell ^{2}(\mathbb{Z})$, where $\unicode[STIX]{x1D706},\unicode[STIX]{x1D703}\in \mathbb{R}$. Avila and Jitomirskaya [The ten Martini problem. Ann. of Math. (2), 170(1) (2009), 303–342] conjectured that, for $2\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6FC}\mathbb{Z}+\mathbb{Z}$, $H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}$ satisfies Anderson localization if $|\unicode[STIX]{x1D706}|>e^{2\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}$. In this paper, we develop a method to treat simultaneous frequency and phase resonances and obtain that, for $2\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6FC}\mathbb{Z}+\mathbb{Z}$, $H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}$ satisfies Anderson localization if $|\unicode[STIX]{x1D706}|>e^{3\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}$.

scholarly journals On the Spectra of Separable 2D Almost Mathieu Operators

2021 ◽  
Author(s):  
Alberto Takase
Keyword(s):  
Positive Measure ◽  
Schrödinger Operators ◽  
Cantor Sets ◽  
Schrodinger Operators ◽  
Discrete Schrödinger Operators ◽  
Mathieu Operator ◽  
Almost Mathieu Operator

AbstractWe consider separable 2D discrete Schrödinger operators generated by 1D almost Mathieu operators. For fixed Diophantine frequencies, we prove that for sufficiently small couplings the spectrum must be an interval. This complements a result by J. Bourgain establishing that for fixed couplings the spectrum has gaps for some (positive measure) Diophantine frequencies. Our result generalizes to separable multidimensional discrete Schrödinger operators generated by 1D quasiperiodic operators whose potential is analytic and whose frequency is Diophantine. The proof is based on the study of the thickness of the spectrum of the almost Mathieu operator and utilizes the Newhouse Gap Lemma on sums of Cantor sets.

scholarly journals Positive Hausdorff Dimensional Spectrum for the Critical Almost Mathieu Operator

2018 ◽  
Vol 368 (1) ◽  
pp. 369-382
Author(s):  
Bernard Helffer ◽  
Qinghui Liu ◽  
Yanhui Qu ◽  
Qi Zhou
Keyword(s):  
Dimensional Spectrum ◽  
Mathieu Operator ◽  
Almost Mathieu Operator

scholarly journals Chebyshev expansion for the component functions of the almost-Mathieu operator

PAMM ◽  
2007 ◽  
Vol 7 (1) ◽  
pp. 2040071-2040072
Author(s):  
Jesús C. Abderramán Marrero
Keyword(s):  
Chebyshev Expansion ◽  
Mathieu Operator ◽  
Almost Mathieu Operator

scholarly journals Dilations of $q$-Commuting Unitaries

2020 ◽  
Author(s):  
Malte Gerhold ◽  
Orr Moshe Shalit
Keyword(s):  
Continuous Function ◽  
Hausdorff Metric ◽  
Natural Structure ◽  
Hölder Continuous ◽  
Lipschitz Continuous ◽  
C Algebras ◽  
Mathieu Operator ◽  
Almost Mathieu Operator ◽  
Continuous Field ◽  
Continuity Of The Spectrum

Abstract Let $q = e^{i \theta } \in \mathbb{T}$ (where $\theta \in \mathbb{R}$), and let $u,v$ be $q$-commuting unitaries, that is, $u$ and $v$ are unitaries such that $vu = quv$. In this paper, we find the optimal constant $c = c_{\theta }$ such that $u,v$ can be dilated to a pair of operators $c U, c V$, where $U$ and $V$ are commuting unitaries. We show that $$\begin{equation*} c_{\theta} = \frac{4}{\|u_{\theta}+u_{\theta}^*+v_{\theta}+v_{\theta}^*\|}, \end{equation*}$$where $u_{\theta }, v_{\theta }$ are the universal $q$-commuting pair of unitaries, and we give numerical estimates for the above quantity. In the course of our proof, we also consider dilating $q$-commuting unitaries to scalar multiples of $q^{\prime}$-commuting unitaries. The techniques that we develop allow us to give new and simple “dilation theoretic” proofs of well-known results regarding the continuity of the field of rotations algebras. In particular, for the so-called “almost Mathieu operator” $h_{\theta } = u_{\theta }+u_{\theta }^*+v_{\theta }+v_{\theta }^*$, we recover the fact that the norm $\|h_{\theta }\|$ is a Lipschitz continuous function of $\theta $, as well as the result that the spectrum $\sigma (h_{\theta })$ is a $\frac{1}{2}$-Hölder continuous function in $\theta $ with respect to the Hausdorff metric. In fact, we obtain this Hölder continuity of the spectrum for every self-adjoint *-polynomial $p(u_{\theta },v_{\theta })$, which in turn endows the rotation algebras with the natural structure of a continuous field of C*-algebras.

scholarly journals On the Green function of the almost-Mathieu operator

2002 ◽  
Vol 35 (10) ◽  
pp. 2449-2455
Author(s):  
Florian G Chmela ◽  
Gustav M Obermair
Keyword(s):  
Green Function ◽  
Mathieu Operator ◽  
Almost Mathieu Operator ◽  
The Green Function

scholarly journals Hofstadter point spectrum trace and the almost Mathieu operator

2018 ◽  
Vol 59 (7) ◽  
pp. 073504 ◽  
Author(s):  
Stéphane Ouvry ◽  
Stephan Wagner ◽  
Shuang Wu
Keyword(s):  
Point Spectrum ◽  
Mathieu Operator ◽  
Almost Mathieu Operator
Sign in / Sign up

Export Citation Format

Share Document