Showing papers on "Almost Mathieu operator published in 2017"

Sharp phase transitions for the almost Mathieu operator

Abstract: It is known that the spectral type of the almost Mathieu operator (AMO) depends in a fundamental way on both the strength of the coupling constant and the arithmetic properties of the frequency. We study the competition between those factors and locate the point where the phase transition from singular continuous spectrum to pure point spectrum takes place, which solves Jitomirskaya’s conjecture. Together with a previous work by Avila, this gives the sharp description of phase transitions for the AMO for the a.e. phase.

Spectral theory of extended Harper’s model and a question by Erdős and Szekeres

Abstract: The extended Harper’s model, proposed by D.J. Thouless in 1983, generalizes the famous almost Mathieu operator, allowing for a wider range of lattice geometries (parametrized by three coupling parameters) by permitting 2D electrons to hop to both nearest and next nearest neighboring (NNN) lattice sites, while still exhibiting its characteristic symmetry (Aubry–Andre duality). Previous understanding of the spectral theory of this model was restricted to two dual regions of the parameter space, one of which is characterized by the positivity of the Lyapunov exponent. In this paper, we complete the picture with a description of the spectral measures over the entire remaining (self-dual) region, for all irrational values of the frequency parameter (the magnetic flux in the model). Most notably, we prove that in the entire interior of this regime, the model exhibits a collapse from purely ac spectrum to purely sc spectrum when the NNN interaction becomes symmetric. In physics literature, extensive numerical analysis had indicated such “spectral collapse,” however so far not even a heuristic argument for this phenomenon could be provided. On the other hand, in the remaining part of the self-dual region, the spectral measures are singular continuous irrespective of such symmetry. The analysis requires some rather delicate number theoretic estimates, which ultimately depend on the solution of a problem posed by Erdős and Szekeres (On the product $$\prod _{k=1}^n(1-z^{a_k})$$ , Publ. de l’Institut mathematique, Paris, 1950).

Singular Continuous Spectrum for Singular Potentials

Abstract: We prove that Schrodinger operators with meromorphic potentials \({(H_{\alpha,\theta}u)_n=u_{n+1}+u_{n-1}+ \frac{g(\theta+n\alpha)}{f(\theta+n\alpha)} u_n}\) have purely singular continuous spectrum on the set \({\{E: L(E) < \delta{(\alpha, \theta)}\}}\), where \({\delta}\) is an explicit function and L is the Lyapunov exponent. This extends results of Jitomirskaya and Liu (Arithmetic spectral transitions for the Maryland model. CPAM, to appear) for the Maryland model and of Avila,You and Zhou (Sharp Phase transitions for the almost Mathieu operator. Preprint, 2015) for the almost Mathieu operator to the general family of meromorphic potentials.

Spectral Characteristics of the Unitary Critical Almost-Mathieu Operator

Abstract: We discuss spectral characteristics of a one-dimensional quantum walk whose coins are distributed quasi-periodically. The unitary update rule of this quantum walk shares many spectral characteristics with the critical Almost-Mathieu Operator; however, it possesses a feature not present in the Almost-Mathieu Operator, namely singularity of the associated cocycles (this feature is, however, present in the so-called Extended Harper’s Model). We show that this operator has empty absolutely continuous spectrum and that the Lyapunov exponent vanishes on the spectrum; hence, this model exhibits Cantor spectrum of zero Lebesgue measure for all irrational frequencies and arbitrary phase, which in physics is known as Hofstadter’s butterfly. In fact, we will show something stronger, namely, that all spectral parameters in the spectrum are of critical type, in the language of Avila’s global theory of analytic quasiperiodic cocycles. We further prove that it has empty point spectrum for each irrational frequency and away from a frequency-dependent set of phases having Lebesgue measure zero. The key ingredients in our proofs are an adaptation of Avila’s Global Theory to the present setting, self-duality via the Fourier transform, and a Johnson-type theorem for singular dynamically defined CMV matrices which characterizes their spectra as the set of spectral parameters at which the associated cocycles fail to admit a dominated splitting.

Cantor spectra of magnetic chain graphs

Abstract: We demonstrate a one-dimensional magnetic system can exhibit a Cantor-type spectrum using an example of a chain graph with δ coupling at the vertices exposed to a magnetic field perpendicular to the graph plane and varying along the chain. If the field grows linearly with an irrational slope, measured in terms of the flux through the loops of the chain, we demonstrate the character of the spectrum relating it to the almost Mathieu operator.

Hofstadter point spectrum trace and the Almost Mathieu operator

Abstract: We consider point spectrum traces in the Hofstadter model. We show how to recover the full quantum Hofstadter trace by integrating these point spectrum traces with the appropriate free density of states on the lattice. This construction is then generalized to the almost Mathieu operator and its n-th moments which can be expressed in terms of generalized Kreft coefficients.