Mathieu Operators Research Articles

On the Spectra of Separable 2D Almost Mathieu Operators

We 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.

Arithmetic version of Anderson localization via reducibility

The arithmetic version of Anderson localization (AL), i.e., AL with explicit arithmetic description on both the localization frequency and the localization phase, was first given by Jitomirskaya (Ann Math 150:1159–1175, 1999) for the almost Mathieu operators (AMO). Later, the result was generalized by Bourgain and Jitomirskaya (Invent Math 148:453–463, 2002) to a class of one dimensional quasi-periodic long-range operators. In this paper, we propose a novel approach based on an arithmetic version of Aubry duality and quantitative reducibility. Our method enables us to prove the same result for the class of quasi-periodic long-range operators in all dimensions, which includes Jitomirskaya (Ann Math 150:1159–1175, 1999) and Bourgain and Jitomirskaya (Invent Math 148:453–463, 2002) as special cases.

Exact dynamical decay rate for the almost Mathieu operator

We prove that the exponential decay rate in expectation is well defined and is equal to the Lyapunov exponent, for supercritical almost Mathieu operators with Diophantine frequencies.

Almost Mathieu operators with completely resonant phases

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 $|\unicode[STIX]{x1D706}|>e^{3\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}$.

Universal hierarchical structure of quasiperiodic eigenfunctions

We determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices of the almost Mathieu operators for all frequencies in the localization regime. This uncovers a universal structure in their behavior, governed by the continued fraction expansion of the frequency, explaining some predictions in physics literature. In addition it proves the arithmetic version of the frequency transition conjecture. Finally, it leads to an explicit description of several non-regularity phenomena in the corresponding non-uniformly hyperbolic cocycles, which is also of interest as both the first natural example of some of those phenomena and, more generally, the first non-artificial model where non-regularity can be explicitly studied.

Spectral gaps of almost Mathieu operators in the exponential regime

For almost Mathieu operator (H_{\lambda,\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+2\lambda \cos2\pi(\theta+n\alpha)u_n, the dry version of the Ten Martini Problem predicts that the spectrum \Sigma_{\lambda,\alpha} of H_{\lambda,\alpha,\theta} has all gaps open for all \lambda\neq 0 and \alpha \in \mathbb{R}\backslash \mathbb{Q} . Avila and Jitomirskaya prove that \Sigma_{\lambda,\alpha} has all gaps open for Diophantine \alpha and 0<|\lambda|<1 . In the present paper, we show that \Sigma_{\lambda,\alpha} has all gaps open for all \alpha \in \mathbb{R}\backslash \mathbb{Q} with small \lambda .

On spectral approximation, Følner sequences and crossed products

In this article we study Følner sequences for operators and mention their relation to spectral approximation problems. We construct a canonical Følner sequence for the crossed product of a discrete amenable group Γ with a concrete C∗-algebra A with a Følner sequence. We also state a compatibility condition for the action of Γ on A. We illustrate our results with two examples: the rotation algebra (which contains interesting operators like almost Mathieu operators or periodic magnetic Schrödinger operators on graphs) and the C∗-algebra generated by bounded Jacobi operators. These examples can be interpreted in the context of crossed products. The crossed products considered can be also seen as a more general frame that included the set of generalized band-dominated operators.

Spectral relationships between kicked Harper and on-resonance double kicked rotor operators

Kicked Harper operators and on-resonance double kicked rotor operators model quantum systems whose semiclassical limits exhibit chaotic dynamics. Recent computational studies indicate a striking resemblance between the spectra of these operators. In this paper we apply C∗-algebra methods to explain this resemblance. We show that each pair of corresponding operators belongs to a common rotation C∗-algebra Bα, prove that their spectra are equal if α is irrational, and prove that the Hausdorff distance between their spectra converges to zero as q increases if α=p/q with p and q coprime integers. Moreover, we show that corresponding operators in Bα are homomorphic images of mother operators in the universal rotation C∗-algebra Aα that are unitarily equivalent and hence have identical spectra. These results extend analogous results for almost Mathieu operators. We also utilize the C∗-algebraic framework to develop efficient algorithms to compute the spectra of these mother operators for rational α and present preliminary numerical results that support the conjecture that their spectra are Cantor sets if α is irrational. This conjecture for almost Mathieu operators, called the ten Martini problem, was recently proven after intensive efforts over several decades. This proof for the almost Mathieu operators utilized transfer matrix methods, which do not exist for the kicked operators. We outline a strategy, based on a special property of loop groups of semisimple Lie groups, to prove this conjecture for the kicked operators.

Norm estimates of almost Mathieu operators

We estimate the norm of the almost Mathieu operator H θ , λ = U θ + U θ * + ( λ / 2 ) ( V θ + V θ * ) , regarded as an element in the rotation C * -algebra A θ = C * ( U θ , V θ unitaries : U θ V θ = e 2 π i θ V θ U θ ) . In the process, we prove for every λ ∈ R and θ ∈ [ 1 4 , 1 2 ] the inequality ∥ H θ , λ ∥ ⩽ 4 + λ 2 - 1 - 1 tan π θ 1 - 1 + cos 2 4 π θ 2 min { 4 , λ 2 } . This significantly improves the inequality ∥ H θ , 2 ∥ ⩽ 2 2 , θ ∈ [ 1 4 , 1 2 ] , conjectured by Béguin, Valette and Zuk.

The spectrum of a class of almost periodic operators

For almost Mathieu operators, it is shown that the occurrence of Cantor spectrum and the existence, for every point in the spectrum and suitable phase parameters, of at least one localized eigenfunction which decays exponentially are inconsistent properties.

Non-invertibility of certain almost Mathieu operators

It is shown that the almost Mathieu operators of the type Te n =e n-1 + λsin(2nr)e n + e n+1 where λ is real and r is a rational multiple of π and {e n :n = 1,2,3,....} an orthonormal basis for a Hilbert space, is notinvertible.

On spectral properties of almost Mathieu operators and connections with irrational rotation $C^*$-algebras

On spectral properties of almost Mathieu operators and connections with irrational rotation $C^*$-algebras

Purely absolutely continuous spectrum for almost Mathieu operators

Using a recent result of Sinai, we prove that the almost Mathieu operators acting onl2(ℤ), (lY,λ Ψ)(n) = Ψ(l+1)+(l−)+λ cos(ωn+α) Ψ(n) have a purely absolutely continuous spectrum for almost all a provided that ω is a good irrational and λ is sufficiently small. Furthermore, the generalized eigen-functions are quasiperiodic.

Almost Mathieu Operators and Rotation C* -Algebras

A concept of eigenstates is discussed for almost Mathieu operators, relating these operators to the rotation C*-algebras they are associated with and to the integrated density of states. For irrational rotations the eigenstates give rise to a notion of multiplicity for all points in the spectrum of an almost Mathieu operator, an integer which is either 1 or 2. It is shown how the occurrence of point spectrum relates to the multiplicity of the corresponding eigenvalues.