Related papers: The Ten Martini Problem

In this paper we use results on reducibility, localization and duality for the Almost Mathieu operator, \[ (H_{b,\phi} x)_n= x_{n+1} +x_{n-1} + b \cos(2 \pi n \omega + \phi)x_n \] on $l^2(\mathbb{Z})$ and its associated eigenvalue equation…

We solve the ten martini problem (Cantor spectrum with no condition on irrational frequencies, previously only established for the almost Mathieu) for a large class of one-frequency quasiperiodic operators, including nonperturbative…

It is shown that for any irrational rotation number and any admissible gap labelling number the almost Mathieu operator (also known as Harper's operator) has a gap in its spectrum with that labelling number. This answers the strong version…

We prove that the spectrum of the almost Mathieu operator is absolutely continuous if and only if the coupling is subcritical. This settles Problem 6 of Barry Simon's list of Schr\"odinger operator problems for the twenty-first century.

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 spectrums of these…

We solve the Dry Ten Martini Problem in the non-critical case, i.e., all possible spectral gaps are open for almost Mathieu operators with $\lambda\ne \pm 1$.

The Dry Ten Martini Problem for Sturmian Hamiltonians is affirmatively solved. Concretely, we prove that all spectral gaps are open for Schr\"odinger operators with Sturmian potentials and non-vanishing coupling constant. A key approach…

I review a recent progress towards solution of the Almost Mathieu equation (A.G. Abanov, J.C. Talstra, P.B. Wiegmann, Nucl. Phys. B 525, 571, 1998), known also as Harper's equation or Azbel-Hofstadter problem. The spectrum of this equation…

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 Ten Martini problem predicts that the spectrum $\Sigma_{\lambda,\alpha}$ of $ H_{\lambda,\alpha,\theta}$…

It is shown that the complete localization of eigenvectors for the almost Mathieu operator entails the absence of Cantor spectrum for this operator.

We solve the Dry Ten Martini Problem for the unitary almost Mathieu operator with Diophantine frequencies in the non-critical regime.

We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches…

In this paper, we add to the characterization of the Fourier spectra for Bernoulli convolution measures. These measures are supported on Cantor subsets of the line. We prove that performing an odd additive translation to half the canonical…

We demonstrate a one-dimensional magnetic system can exhibit a Cantor-type spectrum using an example of a chain graph with $\delta$ coupling at the vertices exposed to a magnetic field perpendicular to the graph plane and varying along the…

We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is non-critical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for…

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice, with the diagonal and the off-diagonal entries given by two sequences generated by the Fibonacci substitution on two letters. We show that the…

For the almost Mathieu operator with a small coupling constant, for a series of spectral gaps, we describe the asymptotic locations of the gaps and get lower bounds for their lengths. The results are obtained by analysing a monodromy…

We study the almost Mathieu operator at critical coupling. We prove that there exists a dense $G_\delta$ set of frequencies for which the spectrum is of zero Hausdorff dimension.

In this paper, we investigate the spectrum of a class of multidimensional quasi-periodic Schr\"odinger operators that exhibit a Cantor spectrum, which provides a resolution to a question posed by Damanik, Fillman, and Gorodetski \cite{DFG}.…

We show that some spectral properties of the almost Mathieu operator with frequency well approximated by rationals can be as poor as at all possible in the class of all one-dimensional discrete Schroedinger operators. For the class of…