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Jul 20, 2013
07/13

by
David Damanik

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We discuss various approaches to localization results for one-dimensional random Schr\"odinger operators, both discrete and continuum. We focus in particular on the approach based on F\"urstenberg's Theorem and the Kunz-Souillard method. These notes are based on a series of five one-hour lectures given at University College London in June/July 2011.

Source: http://arxiv.org/abs/1107.1094v1

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Sep 22, 2013
09/13

by
David Damanik

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These notes are based on a series of six lectures, given during my stay at the CRC 701 in June/July 2008. The lecture series intended to give a survey of some of the results for the almost Mathieu operator that have been obtained since the early 1980's. Specifically, the metal-insulator transition is discussed in detail, along with its relation to the ten Martini problem via duality and reducibility.

Source: http://arxiv.org/abs/0908.1093v1

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Sep 24, 2013
09/13

by
David Damanik

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We show that probability measures on the unit circle associated with Verblunsky coefficients obeying a Coulomb-type decay estimate have no singular continuous component.

Source: http://arxiv.org/abs/math/0412515v1

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Jun 30, 2018
06/18

by
David Damanik

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We consider finite two-way automata and measure the use of two-way motion by counting the number of left moves in accepting computations. Restriction of the automata according to this measure allows us to study in detail the use of two-way motion for the acceptance of regular languages in terms of state complexity. The two-way spectrum of a given regular language is introduced. This quantity reflects the change of size of minimal accepting devices if the use of two-way motion is increased...

Topics: Computing Research Repository, Formal Languages and Automata Theory

Source: http://arxiv.org/abs/1409.5909

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Jun 30, 2018
06/18

by
David Damanik

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In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schr\"odinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an introductory part explaining basic spectral concepts and fundamental results, we present the general theory of such operators, and then provide an overview of known results for specific classes of potentials. Here we focus primarily on the cases of...

Topics: Mathematics, Spectral Theory, Mathematical Physics, Dynamical Systems

Source: http://arxiv.org/abs/1410.2445

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Sep 18, 2013
09/13

by
David Damanik

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The absolutely continuous spectrum of an ergodic family of one-dimensional Schr\"odinger operators is completely determined by the Lyapunov exponent as shown by Ishii, Kotani and Pastur. Moreover, the part of the theory developed by Kotani gives powerful tools for proving the absence of absolutely continuous spectrum, the presence of absolutely continuous spectrum, and even the presence of purely absolutely continuous spectrum. We review these results and their recent applications to a...

Source: http://arxiv.org/abs/math-ph/0605054v1

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Sep 18, 2013
09/13

by
David Damanik

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We consider ergodic families of Schr\"odinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have indeed been established for large classes of operators of this type over the course of the last twenty years. We review the mechanisms leading to these results and briefly discuss analogues for...

Source: http://arxiv.org/abs/math/0509197v1

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Sep 21, 2013
09/13

by
David Damanik

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Following the Killip-Kiselev-Last method, we prove quantum dynamical upper bounds for discrete one-dimensional Schr\"odinger operators with Sturmian potentials. These bounds hold for sufficiently large coupling, almost every rotation number, and every phase.

Source: http://arxiv.org/abs/math-ph/0203018v1

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Sep 19, 2013
09/13

by
David Damanik; Serguei Tcheremchantsev

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We present an approach to quantum dynamical lower bounds for discrete one-dimensional Schr\"odinger operators which is based on power-law bounds on transfer matrices. It suffices to have such bounds for a nonempty set of energies. We apply this result to various models, including the Fibonacci Hamiltonian.

Source: http://arxiv.org/abs/math-ph/0206025v1

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Sep 18, 2013
09/13

by
David Damanik; Michael Goldstein

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We study the quasi-periodic Schr\"odinger equation $$ -\psi"(x) + V(x) \psi(x) = E \psi(x), \qquad x \in \IR $$ in the regime of "small" $V$. Let $(E_m',E"_m)$, $m \in \zv$, be the standard labeled gaps in the spectrum. Our main result says that if $E"_m - E'_m \le \ve \exp(-\kappa_0 |m|)$ for all $m \in \zv$, with $\ve$ being small enough, depending on $\kappa_0 > 0$ and the frequency vector involved, then the Fourier coefficients of $V$ obey $|c(m)| \le...

Source: http://arxiv.org/abs/1209.4331v4

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Sep 21, 2013
09/13

by
David Damanik; Serguei Tcheremchantsev

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We establish quantum dynamical lower bounds for discrete one-dimensional Schr\"odinger operators in situations where, in addition to power-law upper bounds on solutions corresponding to energies in the spectrum, one also has lower bounds following a scaling law. As a consequence, we obtain improved dynamical results for the Fibonacci Hamiltonian and related models.

Source: http://arxiv.org/abs/math-ph/0407017v1

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Sep 18, 2013
09/13

by
David Damanik; Rowan Killip

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We consider Sch\"odinger operators on the half-line, both discrete and continuous, and show that the absence of bound states implies the absence of embedded singular spectrum. More precisely, in the discrete case we prove that if $\Delta + V$ has no spectrum outside of the interval $[-2,2]$, then it has purely absolutely continuous spectrum. In the continuum case we show that if both $-\Delta + V$ and $-\Delta - V$ have no spectrum outside $[0,\infty)$, then both operators are purely...

Source: http://arxiv.org/abs/math-ph/0303001v1

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Sep 21, 2013
09/13

by
Michael Boshernitzan; David Damanik

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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.

Source: http://arxiv.org/abs/0808.2200v2

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Sep 18, 2013
09/13

by
David Damanik; Serguei Naboko

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We consider a class of Jacobi matrices with unbounded coefficients. This class is known to exhibit a first-order phase transition in the sense that, as a parameter is varied, one has purely discrete spectrum below the transition point and purely absolutely continuous spectrum above the transition point. We determine the spectral type and solution asymptotics at the transition point.

Source: http://arxiv.org/abs/math/0605593v1

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Sep 18, 2013
09/13

by
David Damanik; Serguei Tcheremchantsev

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We develop a general method to bound the spreading of an entire wavepacket under Schr\"odinger dynamics from above. This method derives upper bounds on time-averaged moments of the position operator from lower bounds on norms of transfer matrices at complex energies. This general result is applied to the Fibonacci operator. We find that at sufficiently large coupling, all transport exponents take values strictly between zero and one. This is the first rigorous result on anomalous...

Source: http://arxiv.org/abs/math-ph/0502044v1

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Jun 30, 2018
06/18

by
David Damanik; Peter Yuditskii

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The Kotani-Last conjecture states that every ergodic operator in one space dimension with non-empty absolutely continuous spectrum must have almost periodic coefficients. This statement makes sense in a variety of settings; for example, discrete Schr\"odinger operators, Jacobi matrices, CMV matrices, and continuum Schr\"odinger operators. In the main body of this paper we show how to construct counterexamples to the Kotani-Last conjecture for continuum Schr\"odinger operators by...

Topics: Mathematics, Spectral Theory, Mathematical Physics, Dynamical Systems

Source: http://arxiv.org/abs/1405.6343

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Sep 20, 2013
09/13

by
David Damanik; Helge Krueger

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We exhibit examples of almost periodic Verblunsky coefficients for which Herman's subharmonicity argument applies and yields that the associated Lyapunov exponents are uniformly bounded away from zero.

Source: http://arxiv.org/abs/0802.3547v1

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Sep 18, 2013
09/13

by
David Damanik; Rowan Killip

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We study global reflection symmetries of almost periodic functions. In the non-limit periodic case, we establish an upper bound on the Haar measure of the set of those elements in the hull which are almost symmetric about the origin. As an application of this result we prove that in the non-limit periodic case, the criterion of Jitomirskaya and Simon ensuring absence of eigenvalues for almost periodic Schr\"odinger operators is only applicable on a set of zero Haar measure. We complement...

Source: http://arxiv.org/abs/math-ph/0005018v1

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Sep 22, 2013
09/13

by
Artur Avila; David Damanik

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We consider Schr\"odinger operators with ergodic potential $V_\omega(n)=f(T^n(\omega))$, $n \in \Z$, $\omega \in \Omega$, where $T:\Omega \to \Omega$ is a non-periodic homeomorphism. We show that for generic $f \in C(\Omega)$, the spectrum has no absolutely continuous component. The proof is based on approximation by discontinuous potentials which can be treated via Kotani Theory.

Source: http://arxiv.org/abs/math/0409061v1

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Sep 23, 2013
09/13

by
David Damanik; Michael Goldstein

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We consider the KdV equation $$ \partial_t u +\partial^3_x u +u\partial_x u=0 $$ with quasi-periodic initial data whose Fourier coefficients decay exponentially. For any such data and with no limitations on the frequency vector involved (in particular for periodic data), we prove existence and uniqueness in the class of functions which have an expansion with exponentially decaying Fourier coefficients of a solution on a small interval of time, the length of which depends on the given data and...

Source: http://arxiv.org/abs/1212.2674v2

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Jun 28, 2018
06/18

by
David Damanik; Anton Gorodetski

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We generalize the approach to localization in one dimension introduced by Kunz-Souillard, and refined by Delyon-Kunz-Souillard and Simon, in the early 1980's in such a way that certain correlations are allowed. Several applications of this generalized Kunz-Souillard method to almost periodic Schr\"odinger operators are presented. On the one hand, we show that the Schr\"odinger operators on $l^2(\mathbb{Z})$ with limit-periodic potential that have pure point spectrum form a dense...

Topics: Spectral Theory, Mathematics, Mathematical Physics

Source: http://arxiv.org/abs/1510.07349

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Sep 20, 2013
09/13

by
David Damanik; Rowan Killip

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We show that discrete one-dimensional Schr\"odinger operators on the half-line with ergodic potentials generated by the doubling map on the circle, $V_\theta(n) = f(2^n \theta)$, may be realized as the half-line restrictions of a non-deterministic family of whole-line operators. As a consequence, the Lyapunov exponent is almost everywhere positive and the absolutely continuous spectrum is almost surely empty.

Source: http://arxiv.org/abs/math-ph/0405061v1

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Sep 19, 2013
09/13

by
Michael Boshernitzan; David Damanik

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The repetition property of a sequence in a metric space, a notion introduced by us in an earlier paper, is of importance in the spectral analysis of ergodic Schr\"odinger operators. It may be used to exclude eigenvalues for such operators. In this paper we study the question of when a sequence on a torus that is generated by a polynomial or a skew-shift has the repetition property. This provides classes of ergodic Schr\"odinger operators with potentials generated by skew-shifts on...

Source: http://arxiv.org/abs/0708.3234v1

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Sep 18, 2013
09/13

by
David Damanik; Daniel Lenz

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We consider discrete one-dimensional Schr\"odinger operators with Sturmian potentials. For a full-measure set of rotation numbers including the Fibonacci case we prove absence of eigenvalues for all elements in the hull.

Source: http://arxiv.org/abs/math-ph/9903011v1

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Sep 18, 2013
09/13

by
David Damanik; Günter Stolz

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We prove a criterion for absence of eigenvalues for one-dimensional Schr\"odinger operators. This criterion can be regarded as an $L^1$-version of Gordon's theorem and it has a broader range of application. Absence of eigenvalues is then established for quasiperiodic potentials generated by Liouville frequencies and various types of functions such as step functions, H\"older continuous functions and functions with power-type singularities. The proof is based on Gronwall-type a priori...

Source: http://arxiv.org/abs/math-ph/0005015v1

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91

Sep 17, 2013
09/13

by
Artur Avila; David Damanik

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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 almost every phase, settling the measure-theoretical case of Problem 6 of Barry Simon's list of Schr\"odinger operator problems for the twenty-first century.

Source: http://arxiv.org/abs/0711.4291v1

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Sep 24, 2013
09/13

by
David Damanik; Christian Remling

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Consider the Schr\"odinger operators $H_{\pm}=-d^2/dx^2\pm V(x)$. We present a method for estimating the potential in terms of the negative eigenvalues of these operators. Among the applications are inverse Lieb-Thirring inequalities and several sharp results concerning the spectral properties of $H_{\pm}$.

Source: http://arxiv.org/abs/math-ph/0412091v1

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92

Jul 24, 2013
07/13

by
David Damanik; Anton Gorodetski

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We consider Jacobi matrices with zero diagonal and off-diagonals given by elements of the hull of the Fibonacci sequence and show that the spectrum has zero Lebesgue measure and all spectral measures are purely singular continuous. In addition, if the two hopping parameters are distinct but sufficiently close to each other, we show that the spectrum is a dynamically defined Cantor set, which has a variety of consequences for its local and global fractal dimension.

Source: http://arxiv.org/abs/0807.3024v2

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Sep 18, 2013
09/13

by
David Damanik; Barry Simon

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We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szeg\H{o} asymptotics off the real axis. A key idea is to prove the equivalence of Szeg\H{o} asymptotics and of Jost asymptotics for the Jost solution. We also prove $L^2$ convergence of Szeg\H{o} asymptotics on the spectrum.

Source: http://arxiv.org/abs/math/0502486v1

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Jun 29, 2018
06/18

by
David Damanik; Anton Gorodetski

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We show that under natural technical conditions, the sum of a $C^2$ dynamically defined Cantor set with a compact set in most cases (for almost all parameters) has positive Lebesgue measure, provided that the sum of the Hausdorff dimensions of these sets exceeds one. As an application, we show that for many parameters, the Square Fibonacci Hamiltonian has spectrum of positive Lebesgue measure, while at the same time the density of states measure is purely singular.

Topics: Spectral Theory, Dynamical Systems, Mathematical Physics, Mathematics

Source: http://arxiv.org/abs/1601.01639

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40

Sep 17, 2013
09/13

by
David Damanik; Günter Stolz

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We consider continuum one-dimensional Schr\"odinger operators with potentials that are given by a sum of a suitable background potential and an Anderson-type potential whose single-site distribution has a continuous and compactly supported density. We prove exponential decay of the expectation of the finite volume correlators, uniform in any compact energy region, and deduce from this dynamical and spectral localization. The proofs implement a continuum analog of the method Kunz and...

Source: http://arxiv.org/abs/0912.3568v1

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Sep 18, 2013
09/13

by
David Damanik; Daniel Lenz

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We consider discrete one-dimensional Schr\"odinger operators with quasi-Sturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, it is shown that the operators have purely singular continuous spectrum and their spectrum is a Cantor set of Lebesgue measure zero. We also exhibit a subclass having purely $\alpha$-continuous...

Source: http://arxiv.org/abs/math-ph/0105034v1

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Sep 23, 2013
09/13

by
David Damanik; Serguei Tcheremchantsev

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We develop further the approach to upper and lower bounds in quantum dynamics via complex analysis methods which was introduced by us in a sequence of earlier papers. Here we derive upper bounds for non-time averaged outside probabilities and moments of the position operator from lower bounds for transfer matrices at complex energies. Moreover, for the time-averaged transport exponents, we present improved lower bounds in the special case of the Fibonacci Hamiltonian. These bounds lead to an...

Source: http://arxiv.org/abs/0801.3399v1

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44

Sep 18, 2013
09/13

by
David Damanik; Serguei Tcheremchantsev

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We consider transport exponents associated with the dynamics of a wavepacket in a discrete one-dimensional quantum system and develop a general method for proving upper bounds for these exponents in terms of the norms of transfer matrices at complex energies. Using this method, we prove such upper bounds for the Fibonacci Hamiltonian. Together with the known lower bounds, this shows that these exponents are strictly between zero and one for sufficiently large coupling and the large coupling...

Source: http://arxiv.org/abs/cond-mat/0502004v1

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Sep 18, 2013
09/13

by
David Damanik; Daniel Lenz

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This paper is concerned with the concept of linear repetitivity in the theory of tilings. We prove a general uniform subadditive ergodic theorem for linearly repetitive tilings. This theorem unifies and extends various known (sub)additive ergodic theorems on tilings. The results of this paper can be applied in the study of both random operators and lattice gas models on tilings.

Source: http://arxiv.org/abs/math/0005062v1

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Sep 18, 2013
09/13

by
David Damanik; Anton Gorodetski

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We consider the trace map associated with the Fibonacci Hamiltonian as a diffeomorphism on the invariant surface associated with a given coupling constant and prove that the non-wandering set of this map is hyperbolic if the coupling is sufficiently small. As a consequence, for these values of the coupling constant, the local and global Hausdorff dimension and the local and global box counting dimension of the spectrum of the Fibonacci Hamiltonian all coincide and are smooth functions of the...

Source: http://arxiv.org/abs/0806.0645v1

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Sep 18, 2013
09/13

by
David Damanik; Gerald Teschl

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We consider discrete one-dimensional Schroedinger operators whose potentials decay asymptotically like an inverse square. In the super-critical case, where there are infinitely many discrete eigenvalues, we compute precise asymptotics of the number of eigenvalues below a given energy E as this energy tends to the bottom of the essential spectrum.

Source: http://arxiv.org/abs/math/0509110v1

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Sep 22, 2013
09/13

by
David Damanik; Anton Gorodetski

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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 prove that the thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. We also show that at small coupling, all gaps allowed by the gap labeling theorem are...

Source: http://arxiv.org/abs/1001.2552v1

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Sep 18, 2013
09/13

by
David Damanik; Barry Simon

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We present necessary and sufficient conditions on the Jost function for the corresponding Jacobi parameters $a_n -1$ and $b_n$ to have a given degree of exponential decay.

Source: http://arxiv.org/abs/math/0502487v1

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Sep 20, 2013
09/13

by
David Damanik; Zheng Gan

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We exhibit d-dimensional limit-periodic Schrodinger operators that are uniformly localized in the strongest sense possible. That is, for each of these operators, there is a uniform exponential decay rate such that every element of the hull of the corresponding Schrodinger operator has a complete set of eigenvectors that decay exponentially off their centers of localization at least as fast as prescribed by the uniform decay rate. Consequently, these operators exhibit uniform dynamical...

Source: http://arxiv.org/abs/1207.5881v1

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Sep 17, 2013
09/13

by
David Damanik; Zheng Gan

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We exhibit limit-periodic Schr\"odinger operators that are uniformly localized in the strongest sense possible. That is, for these operators there are uniform exponential decay rates such that every element of the hull has a complete set of eigenvectors that decay exponentially off their centers of localization at least as fast as prescribed by the uniform decay rate. Consequently, these operators exhibit uniform dynamical localization.

Source: http://arxiv.org/abs/1003.1695v3

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Sep 22, 2013
09/13

by
David Damanik; Zheng Gan

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We investigate the spectral properties of the discrete one-dimensional Schr\"odinger operators whose potentials are generated by continuous sampling along the orbits of a minimal translation of a Cantor group. We show that for given Cantor group and minimal translation, there is a dense set of continuous sampling functions such that the spectrum of the associated operators has zero Hausdorff dimension and all spectral measures are purely singular continuous. The associated Lyapunov...

Source: http://arxiv.org/abs/0906.3340v3

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Sep 21, 2013
09/13

by
David Damanik; Anton Gorodetski

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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...

Source: http://arxiv.org/abs/0901.4383v1

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Sep 22, 2013
09/13

by
David Damanik; Michael Landrigan

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We consider discrete one-dimensional Schr\"odinger operators on the whole line and establish a criterion for continuity of spectral measures with respect to $\log$-Hausdorff measures. We apply this result to operators with Sturmian potentials and thereby prove logarithmic quantum dynamical lower bounds for all coupling constants and almost all rotation numbers, uniformly in the phase.

Source: http://arxiv.org/abs/math-ph/0201009v1

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Sep 21, 2013
09/13

by
David Damanik; Rowan Killip

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We prove absence of absolutely continuous spectrum for discrete one-dimensional Schr\"odinger operators on the whole line with certain ergodic potentials, $V_\omega(n) = f(T^n(\omega))$, where $T$ is an ergodic transformation acting on a space $\Omega$ and $f: \Omega \to \R$. The key hypothesis, however, is that $f$ is discontinuous. In particular, we are able to settle a conjecture of Aubry and Jitomirskaya--Mandel'shtam regarding potentials generated by irrational rotations on the torus....

Source: http://arxiv.org/abs/math-ph/0402070v1

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Jun 30, 2018
06/18

by
David Damanik; Anton Gorodetski

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We consider the spectrum of discrete Schr\"odinger operators with Sturmian potentials and show that for sufficiently large coupling, its Hausdorff dimension and its upper box counting dimension are the same for Lebesgue almost every value of the frequency.

Topics: Mathematics, Spectral Theory, Mathematical Physics, Dynamical Systems

Source: http://arxiv.org/abs/1406.4810

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Sep 19, 2013
09/13

by
Michael Boshernitzan; David Damanik

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We consider discrete Schr"odinger operators on the line with potentials generated by a minimal homeomorphism on a compact metric space and a continuous sampling function. We introduce the concepts of topological and metric repetition property. Assuming that the underlying dynamical system satisfies one of these repetition properties, we show using Gordon's Lemma that for a generic continuous sampling function, the associated Schr"odinger operators have no eigenvalues in a topological...

Source: http://arxiv.org/abs/0708.1263v1

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Sep 22, 2013
09/13

by
David Damanik; Zheng Gan

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We investigate the spectral properties of Schr\"odinger 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 a Cantor set of positive Lebesgue measure and...

Source: http://arxiv.org/abs/0906.3337v3

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Sep 21, 2013
09/13

by
David Damanik; Daniel Lenz

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In this paper we introduce a method that allows one to prove uniform local results for one-dimensional discrete Schr\"odinger operators with Sturmian potentials. We apply this method to the transfer matrices in order to study the Lyapunov exponent and the growth rate of eigenfunctions. This gives uniform vanishing of the Lyapunov exponent on the spectrum for all irrational rotation numbers. For irrational rotation numbers with bounded continued fraction expansion, it gives uniform...

Source: http://arxiv.org/abs/math-ph/9905008v1

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Sep 22, 2013
09/13

by
David Damanik; Daniel Lenz

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This paper is concerned with uniform convergence in the multiplicative ergodic theorem on aperiodic subshifts. If such a subshift satisfies a certain condition, originally introduced by Boshernitzan, every locally constant SL(2,R)-valued cocycle is uniform. As a consequence, the corresponding Schr\"odinger operators exhibit Cantor spectrum of Lebesgue measure zero. An investigation of Boshernitzan's condition then shows that these results cover all earlier results of this type and,...

Source: http://arxiv.org/abs/math/0403190v1