The spectrum of the weakly coupled Fibonacci Hamiltonian
David Damanik Anton Gorodetski
Electronic Research Announcements 2009, 16(0): 23-29 doi: 10.3934/era.2009.16.23
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 zero, of its thickness and its Hausdorff dimension. We announce the following results and explain some key ideas that go into their proofs. The thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. Moreover, the length of every gap tends to zero linearly. Finally, for sufficiently small coupling, the sum of the spectrum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by Even-Dar Mandel and Lifshitz.
keywords: Fibonacci Hamiltonian Hausdorff dimension Spectrum trace map thickness. dynamically defined Cantor set
Characterizations of uniform hyperbolicity and spectra of CMV matrices
David Damanik Jake Fillman Milivoje Lukic William Yessen
Discrete & Continuous Dynamical Systems - S 2016, 9(4): 1009-1023 doi: 10.3934/dcdss.2016039
We provide an elementary proof of the equivalence of various notions of uniform hyperbolicity for a class of GL$(2,\mathbb{C})$ cocycles and establish a Johnson-type theorem for extended CMV matrices, relating the spectrum to the set of points on the unit circle for which the associated Szegő cocycle is not uniformly hyperbolic.
keywords: Linear cocycles generalized eigenfunctions CMV matrices uniform hyperbolicity orthogonal polynomials.
Pinned repetitions in symbolic flows: preliminary results
Michael Boshernitzan David Damanik
Conference Publications 2009, 2009(Special): 869-878 doi: 10.3934/proc.2009.2009.869
We consider symbolic flows over finite alphabets and study certain kinds of repetitions in these sequences. Positive and negative results for the existence of such repetitions are given for codings of interval exchange transformations and codings of quadratic polynomials.
keywords: symbolic flows repetitions subshifts interval exchange transformations powers
A general description of quantum dynamical spreading over an orthonormal basis and applications to Schrödinger operators
David Damanik Serguei Tcheremchantsev
Discrete & Continuous Dynamical Systems - A 2010, 28(4): 1381-1412 doi: 10.3934/dcds.2010.28.1381
We discuss the long-time behavior of solutions to the Schrödinger equation in some separable Hilbert space, with particular emphasis on the spreading over some orthonormal basis. Various ways of studying wavepacket spreading from this perspective are described and their inter-relations investigated. We also state and discuss known results for concrete quantum systems relative to this general framework.
keywords: Schrödinger equation Schrödinger operators. Quantum dynamics
Spectral properties of limit-periodic Schrödinger operators
David Damanik Zheng Gan
Communications on Pure & Applied Analysis 2011, 10(3): 859-871 doi: 10.3934/cpaa.2011.10.859
We investigate the spectral properties of Schrödinger operators in $l^2(Z)$ with limit-periodic potentials. The perspective we take was recently proposed by Avila and is based on regarding such potentials as generated by continuous sampling along the orbits of a minimal translation of a Cantor group. This point of view allows one to separate the base dynamics and the sampling function. We show that for any such base dynamics, the spectrum is of positive Lebesgue measure and purely absolutely continuous for a dense set of sampling functions, and it is of zero Lebesgue measure and purely singular continuous for a dense $G_\delta$ set of sampling functions.
keywords: limit-periodic potentials. Schrödinger operators
Schrödinger operators defined by interval-exchange transformations
Jon Chaika David Damanik Helge Krüger
Journal of Modern Dynamics 2009, 3(2): 253-270 doi: 10.3934/jmd.2009.3.253
We discuss discrete one-dimensional Schrödinger operators whose potentials are generated by an invertible ergodic transformation of a compact metric space and a continuous real-valued sampling function. We pay particular attention to the case where the transformation is a minimal interval-exchange transformation. Results about the spectral type of these operators are established. In particular, we provide the first examples of transformations for which the associated Schrödinger operators have purely singular spectrum for every nonconstant continuous sampling function.
keywords: continuous spectrum singular spectrum interval-exchange transformations. Ergodic Schrödinger operators

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