Browsing by Author "Damanik, David"
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Item Absence of absolutely continuous spectrum for generic quasi-periodic Schrödinger operators on the real line(Springer Nature, 2022) Damanik, David; Lenz, DanielWe show that a generic quasi-periodic Schrödinger operator in L2(ℝ) has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schrödinger operator with the resulting potential has empty absolutely continuous spectrum.Item Absolutely continuous spectrum for CMV matrices with small quasi-periodic Verblunsky coefficients(American Mathematical Society, 2022) Li, Long; Damanik, David; Zhou, QiWe consider standard and extended CMV matrices with small quasi-periodic Verblunsky coefficients and show that on their essential spectrum, all spectral measures are purely absolutely continuous. This answers a question of Barry Simon from 2005.Item Anderson localization for quasi-periodic CMV matrices and quantum walks(Elsevier, 2019) Wang, Fengpeng; Damanik, DavidWe consider CMV matrices, both standard and extended, with analytic quasi-periodic Verblunsky coefficients and prove Anderson localization in the regime of positive Lyapunov exponents. This establishes the CMV analog of a result Bourgain and Goldstein proved for discrete one-dimensional Schrödinger operators. We also prove a similar result for quantum walks on the integer lattice with suitable analytic quasi-periodic coins.Item Anderson localization for radial tree graphs with random branching numbers(Elsevier, 2019) Damanik, David; Sukhtaiev, SelimWe prove Anderson localization for the discrete Laplace operator on radial tree graphs with random branching numbers. Our method relies on the representation of the Laplace operator as the direct sum of half-lineﾠJacobi matricesﾠwhose entries are non-degenerate, independent, identically distributed random variables with singular distributions.Item Cantor spectrum of CMV matrices, Jacobi matrices and Schrodinger operators with dynamically defined coefficients and potentials(2020-04-22) Jun, Hyunkyu; Damanik, DavidIn this thesis, we consider CMV matrices, Jacobi matrices and Schr\"{o}dinger operators while assuming that the coefficients and potentials are generated by dynamical systems. One of the major parts investigates continuous cocycles arising from CMV and Jacobi matrices. Assuming the Verblunsky and Jacobi coefficients arise from generalized skew-shifts, we prove that uniform hyperbolicity of the associated cocycles is $C^0$-dense. This implies that the associated CMV and Jacobi matrices have Cantor spectrum for a generic continuous sampling map. Another major part concerns the Fibonacci Hamiltonian. In the classical Fibonacci Hamiltonian, a sampling function is a locally constant function of a very special form. In this thesis, we study whether spectral results of the classical Fibonacci Hamiltonian can be extended to more general sampling functions. We provide the trace map description of the spectrum and extend the results for the classical Fibonacci Hamiltonian to arbitrary locally constant sampling functions.Item Direct and Inverse Spectral Theory for the Hamiltonian System with Measure Coecients(2023-04-19) Wang, Chunyi; Damanik, DavidThis thesis discusses the direct and inverse spectral theory of Hamiltonian systems with measure coefficients, which can cover more singular cases. In the first part, we define self-adjoint relations associated with the systems and develop Weyl-Titchmarsh theory for these relations. Then, we develop subordinacy theory for the relations and discuss several cases when the absolutely continuous spectrum appears. Finally, we develop inverse uniqueness results for Hamiltonian systems with measure coefficients by applying de Branges’ subspace ordering theorem. Overall, this thesis contributes to the study of Hamiltonian systems with measure coefficients, expands the self-adjoint operator theory to a more general class of physical models, and investigates common spectral properties among different models.Item Ergodic Schrödinger operators in the infinite measure setting(EMS Press, 2021) Boshernitzan, Michael; Damanik, David; Fillman, Jake; Lukic, MilivojeWe develop the basic theory of ergodic Schrödinger operators, which is well known for ergodic probability measures, in the case of a base dynamics on an infinite measure space. This includes the almost sure constancy of the spectrum and the spectral type, the definition and discussion of the density of states measure and the Lyapunov exponent, as well as a version of the Pastur–Ishii theorem. We also give some counterexamples that demonstrate that some results do not extend from the finite measure case to the infinite measure case. These examples are based on some constructions in infinite ergodic theory that may be of independent interest.Item Generic spectral results for CMV matrices with dynamically defined Verblunsky coefficients(Elsevier, 2020) Fang, Licheng; Damanik, David; Guo, ShuzhengWe consider CMV matrices with dynamically defined Verblunsky coefficients. These coefficients are obtained by continuous sampling along the orbits of an ergodic transformation. We investigate whether certain spectral phenomena are generic in the sense that for a fixed base transformation, the set of continuous sampling functions for which the spectral phenomenon occurs is residual. Among the phenomena we discuss are the absence of absolutely continuous spectrum and the vanishing of the Lebesgue measure of the spectrum.Item Improved Spectral Calculations for Discrete Schroedinger Operators(2013-09-16) Puelz, Charles; Embree, Mark; Sorensen, Danny C.; Damanik, DavidThis work details an O(n^2) algorithm for computing the spectra of discrete Schroedinger operators with periodic potentials. Spectra of these objects enhance our understanding of fundamental aperiodic physical systems and contain rich theoretical structure of interest to the mathematical community. Previous work on the Harper model led to an O(n^2) algorithm relying on properties not satisfied by other aperiodic operators. Physicists working with the Fibonacci Hamiltonian, a popular quasicrystal model, have instead used a problematic dynamical map approach or a sluggish O(n^3) procedure for their calculations. The algorithm presented in this work, a blend of well-established eigenvalue/vector algorithms, provides researchers with a more robust computational tool of general utility. Application to the Fibonacci Hamiltonian in the sparsely studied intermediate coupling regime reveals structure in canonical coverings of the spectrum that will prove useful in motivating conjectures regarding band combinatorics and fractal dimensions.Item Isospectral dynamics of reflectionless Jacobi operators(2018-04-02) VandenBoom, Tom; Damanik, DavidThis thesis focuses on the isospectral torus of reflectionless Jacobi operators and the dynamics of its automorphisms. The novel perspective which it hopes to advertise is one of the joint utility of inverse spectral theoretic techniques and dynamical techniques to address direct and inverse spectral problems, respectively. Concretely, we use the former perspective to prove the reducibility of the shift cocycle for certain reflectionless Jacobi operators, and we use the latter perspective to prove spectral atypicality of discrete Schroedinger operators.Item Johnson–Schwartzman gap labelling for ergodic Jacobi matrices(EMS Press, 2023) Damanik, David; Fillman, Jake; Zhang, ZhengheWe consider two-sided Jacobi matrices whose coefficients are obtained by continuous sampling along the orbits of a homeomorphism on a compact metric space. Given an ergodic probability measure, we study the topological structure of the associated almost sure spectrum. We establish a gap labelling theorem in the spirit of Johnson and Schwartzman. That is, we show that the constant value the integrated density of states takes in a gap of the spectrum must belong to the countable Schwartzman group of the base dynamics. This result is a natural companion to a recent result of Alkorn and Zhang, which established a Johnson-type theorem for the families of Jacobi matrices in question.Item Limit-periodic Schrödinger operators with Lipschitz continuous IDS(American Mathematical Society, 2019) Damanik, David; Fillman, JakeWe show that there exist limit-periodic Schrödinger operators such that the associated integrated density of states is Lipschitz continuous. These operators arise in the inverse spectral theoretic KAM approach of Pöschel.Item Multidimensional Almost-Periodic Schrödinger Operators with Cantor Spectrum(Springer, 2019) Damanik, David; Fillman, Jake; Gorodetski, AntonWe construct multidimensional almost-periodic Schrödinger operators whose spectrum has zero lower box-counting dimension. In particular, the spectrum in these cases is a generalized Cantor set of zero Lebesgue measure.Item Multidimensional Schrödinger operators whose spectrum features a half-line and a Cantor set(Elsevier, 2021) Damanik, David; Fillman, Jake; Gorodetski, AntonWe construct multidimensional Schrödinger operators with a spectrum that has no gaps at high energies and that is nowhere dense at low energies. This gives the first example for which this widely expected topological structure of the spectrum in the class of uniformly recurrent Schrödinger operators, namely the coexistence of a half-line and a Cantor-type structure, can be confirmed. Our construction uses Schrödinger operators with separable potentials that decompose into one-dimensional potentials generated by the Fibonacci sequence and relies on the study of such operators via the trace map and the Fricke-Vogt invariant. To show that the spectrum contains a half-line, we prove an abstract Bethe–Sommerfeld criterion for sums of Cantor sets which may be of independent interest.Item Must the Spectrum of a Random Schrödinger Operator Contain an Interval?(Springer Nature, 2022) Damanik, David; Gorodetski, AntonWe consider Schrödinger operators in ℓ2(Z) whose potentials are given by independent (not necessarily identically distributed) random variables. We ask whether it is true that almost surely its spectrum contains an interval. We provide an affirmative answer in the case of random potentials given by a sum of a perturbatively small quasi-periodic potential with analytic sampling function and Diophantine frequency vector and a term of Anderson type, given by independent identically distributed random variables (with some small-gap assumption for the support of the single-site distribution). The proof proceeds by extending a result about the presence of ground states for atypical realizations of the classical Anderson model, which we prove here as well and which appears to be new.Item New Anomalous Lieb-Robinson Bounds in Quasiperiodic XY Chains(American Physical Society, 2014) Damanik, David; Lemm, Marius; Lukic, Milivoje; Yessen, WilliamWe announce and sketch the rigorous proof of a new kind of anomalous (or sub-ballistic) Lieb-Robinson (LR) bound for an isotropic XY chain in a quasiperiodic transversal magnetic field. Instead of the usual effective light cone |x|≤v|t|, we obtain |x|≤v|t|α for some 0<α<1. We can characterize the allowed values of α exactly as those exceeding the upper transport exponent α+u of a one-body Schrödinger operator. To our knowledge, this is the first rigorous derivation of anomalous quantum many-body transport. We also discuss anomalous LR bounds with power-law tails for a random dimer field.Item New sufficient condition for Hamiltonian paths(2008) Jennings, Landon; Damanik, DavidThis paper proves a sufficient condition for the existence of Hamiltonian paths in simple connected graphs. This condition was conjectured in 2006 by a computer program named Graffiti.pc. Given examples will show that this new condition detects Hamiltonian paths that a theorem by Chvatal does not. A second condition conjectured by Graffiti.pc is shown to satisfy Chvatal's condition. Thus this second conjecture, while true, does not improve on known results.Item On the existence and uniqueness of global solutions for the KdV equation with quasi-periodic initial data(American Mathematical Society, 2015) Damanik, David; Goldstein, MichaelWe consider the KdV equation ∂tu+∂3xu+u∂xu=0 with quasi-periodic initial data whose Fourier coefficients decay exponentially and 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 the frequency vector involved. For a Diophantine frequency vector and for small quasi-periodic data (i.e., when the Fourier coefficients obey |c(m)|≤εexp(−κ0|m|) with ε>0 sufficiently small, depending on κ0>0 and the frequency vector), we prove global existence and uniqueness of the solution. The latter result relies on our recent work [Publ. Math. Inst. Hautes Études Sci. 119 (2014) 217] on the inverse spectral problem for the quasi-periodic Schrӧdinger equation.Item Open Problems and Conjectures Related to the Theory of Mathematical Quasicrystals(Springer, 2016) Adiceam, Faustin; Damanik, David; Gähler, Franz; Grimm, Uwe; Haynes, Alan; Julien, Antoine; Navas, Andrés; Sadun, Lorenzo; Weiss, BarakThis list of problems arose as a collaborative effort among the participants of the Arbeitsgemeinschaft on Mathematical Quasicrystals, which was held at the Mathematisches Forschungsinstitut Oberwolfach in October 2015. The purpose of our meeting was to bring together researchers from a variety of disciplines, with a common goal of understanding different viewpoints and approaches surrounding the theory of mathematical quasicrystals. The problems below reflect this goal and this diversity and we hope that they will motivate further cross-disciplinary research and lead to new advances in our overall vision of this rapidly developing field.Item Opening gaps in the spectrum of strictly ergodic Schrodinger operators(European Mathematical Society, 2012) Avila, Artur; Bochi, Jairo; Damanik, DavidWe consider Schrodinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the possible gaps in the spectrum can be canonically labelled by an at most countable set defined purely in terms of the dynamics. Which labels actually appear depends on the choice of the sampling function; the missing labels are said to correspond to collapsed gaps. Here we show that for any collapsed gap, the sampling function may be continuously deformed so that the gap immediately opens. As a corollary, we conclude that for generic sampling functions, all gaps are open. The proof is based on the analysis of continuous SL.2;R/ cocycles, for which we obtain dynamical results of independent interest.