We study Hausdorff-dimensional spectral properties of certain “whole-line” quasiperiodic discrete Schrödinger operators by
using the extension of the Gilbert–Pearson subordinacy theory that we previously developed in [19].
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... Intuitively, Hausdorff dimension is associated with lim inf 1 of the power-law behavior of the concentration function of the measure. The zero upper bound, using the techniques of [18,23], can be achieved through controlled occasional largeness of transfer matrices, which, not surprisingly, does happen in the regime of positive Lyapunov exponents. ...
... Power-law subordinacy theory [18,23] has linked 3 certain properties of formal solutions to Hu = Eu of general one-dimensional operators (6) (Hu)(n) = u(n + 1) + u(n − 1) + v n u(n), ...
... It is well known (c.f. (2.5) in [23] or (1.27) in [33]) that ...
We develop tools to study arithmetically induced singular continuous spectrum in the neighborhood of the arithmetic transition in the hyperbolic regime. This leads to first transition-capturing upper bounds on packing and multifractal dimensions of spectral measures. We achieve it through the proof of partial localization of generalized eigenfunctions, another first result of its kind in the singular continuous regime. The proof is based also on a general criterion for lower bounds on concentrations of Borel measures as a corollary of boundary behavior of their Borel-type transforms, that may be of wider use and independent interest.
... The following facts are well known in the past literatures(see e.g. section 3, [25]). For any ϕ ∈ (−π/2, π/2], ...
... In view of (2.10), a direct consequence of (3.26) is (e.g. Lemma 5 in [25]): ...
... Let q n be given as in the continued fraction approximants to α, see (2.8). The following lemma about the ergodicity of an irrational rotation can be found e.g. in [25]. Lemma 6.3 (Lemma 9, [25]). ...
We study fractal dimension properties of singular Jacobi operators. We prove quantitative lower spectral/quantum dynamical bounds for general operators with strong repetition properties and controlled singularities. For analytic quasiperiodic Jacobi operators in the positive Lyapunov exponent regime, we obtain a sharp arithmetic criterion of full spectral dimensionality. The applications include the extended Harper's model where we obtain arithmetic results on spectral dimensions and quantum dynamical exponents.
... Properties of spectral measures and eigenfunctions can be studied for concrete models of interest and the above discussed bounds have been used to obtain dynamical results in cases where there is currently no alternative approach to study the dynamics. In particular, this approach has been used to obtain dynamical results for a number of quasiperiodic operators [11,13,23] and for operators with decaying potentials [25]. ...
... It is known [6,38,39] that for every λ > 0, H λ has purely singular continuous spectrum, and moreover, its spectrum (as a set) is a Cantor set of zero Lebesgue measure. Lower bounds on wavepacket spreading rates for H λ were recently shown in [23]. We are going to show both upper and lower bounds for the dynamics of H λ which imply that the spreading rate is intermediate between ballistic (∼ T at time T ) and localized (∼ T 0 ). ...
... It remains to prove the second part of Theorem 1.6, involving the lower bound on dynamics. The lower bound on dynamics for the Fibonacci Hamiltonian has been proved recently in [23]. We present a sketch of the argument here, making explicit the behavior of the bound in the large coupling regime. ...
We derive a general upper bound on the spreading rate of wavepackets in the framework of Schr\"odinger time evolution. Our result consists of showing that a portion of the wavepacket cannot escape outside a ball whose size grows dynamically in time, where the rate of this growth is determined by properties of the spectral measure and by spatial properties of solutions of an associated time independent Schr\"odinger equation. We also derive a new lower bound on the spreading rate, which is strongly connected with our upper bound. We apply these new bounds to the Fibonacci Hamiltonian--the most studied one-dimensional model of quasicrystals. As a result, we obtain for this model upper and lower dynamical bounds establishing wavepacket spreading rates which are intermediate between ballistic transport and localization. The bounds have the same qualitative behavior in the limit of large coupling.
... They can be divided into two classes • Those that hold for all frequencies (e.g. [38,12,25,41,42,43]) ...
... However Hausdorff dimension is a poor tool for characterizing the singular continuous spectral measures arising in the regime of positive Lyapunov exponents, as it is always equal to zero (for a.e. phase for any ergodic case [57], and for every phase for one frequency analytic potentials [38] 3 ). Similarly, the lower transport exponent is always zero for piecewise Lipshitz potentials [25,43]. ...
... In some sense [12] is a result of this type. 3 The result of [38] is formulated for trigonometric polynomial v. However it extends to the analytic case -and more -by the method of [43]. ...
We introduce a notion of -almost periodicity and prove quantitative lower spectral/quantum dynamical bounds for general bounded -almost periodic potentials. Applications include a sharp arithmetic criterion of full spectral dimensionality for analytic quasiperiodic Schr\"odinger operators in the positive Lyapunov exponent regime and arithmetic criteria for families with zero Lyapunov exponents, with applications to Sturmian potentials and the critical almost Mathieu operator.
... More precisely, even though in this case there is no uniform exponential decay of the generalized eigenfunctions, there are large intervals on which solutions have a localized behavior. The existence of such intervals leads to certain estimates of restricted ℓ 2 -norms of generalized eigenfunctions (see Proposition 3.10), which are known to be connected to fractal continuity properties of the spectral measures [17,18,5]. ...
... Subordinacy [9,25] and power-law subordinacy [5,17,18] relate asymptotic properties of the L-norms of solutions to (2.5) to the boundary behavior of the Borel transform of µ. For every L > 0, let ...
Let H be a quasiperiodic Schr\"{o}dinger operator generated by a monotone potential, as defined in [16]. Following [20], we study the connection between the Lyapunov exponent , arithmetic properties of the frequency , and certain fractal-dimensional properties of the spectral measures of H.
... Variation of parameters is a standard technique in ODE theory and used as an especially powerful tool in spectral theory by Gilbert-Pearson [38] and in Jacobi matrix spectral theory by Khan-Pearson [49]. It was then developed by Jitomirskaya-Last [44,45,46] and Killip-Kiselev-Last [50], from which we take Proposition 21.1. ...
... Here q n are the second kind polynomials defined in Section 8. For (21.1), see [44,45,46,50]. This immediately implies: Corollary 21.2 (Avila-Last-Simon [6]). ...
A review of the uses of the CD kernel in the spectral theory of orthogonal polynomials, concentrating on recent results.
... θ ∈ M (the latter is known for general ergodic potentials [26]). The statement for all θ has only been known for irrational rotations of T 1 (proved for trigonometric polynomials in [14], and follows easily for piecewise functions from the results of [15]). ...
... Then it would take a point on T at most q m + q m−1 steps (under the α1 h2,n -rotation) to enter each interval of length 1 |h1,n| √ h 2 1,n +h 2 2,n on T (e.g. [14]), which means it would take a point on l 2 (t) at most q m + q m−1 − 1 steps (under the √ h 2 1,n +h 2 2,n α1 |h2,n| -rotation) to enter each interval of length 1 |h1,n| = r n on the graph of l 2 (t). Moreover, it is easy to see that the distance from any x ∈ T 2 to l 2 (t) is bounded by 1 √ h 2 1,n +h 2 2,n < r n . ...
In this paper we obtain upper quantum dynamical bounds as a corollary of positive Lyapunov exponent for Schr\"odinger operators , where is a piecewise H\"older function on a compact Riemannian manifold , and is a uniquely ergodic volume preserving map with zero topological entropy. As corollaries we obtain localization-type statements for shifts and skew-shifts on higher dimensional tori with arithmetic conditions on the parameters. These are the first localization-type results with precise arithmetic conditions for multi-frequency quasiperiodic and skew-shift potentials.
... We refer the reader to [2,3,15] for subsequent developments. Apart from the implications for dynamics, the notion of α-continuity has the advantage of being accessible by an investigation of solutions to (2) as was realized by Jitomirskaya and Last [20,21,22]. In these papers they establish an extension of the classical Gilbert-Pearson theory of subordinacy [11,12,24] and present several applications. ...
We study the spectral properties of discrete one-dimensional Schr\"odinger operators with Sturmian potentials. It is shown that the point spectrum is always empty. Moreover, for rotation numbers with bounded density, we establish purely -continuous spectrum, uniformly for all phases. The proofs rely on the unique decomposition property of Sturmian potentials, a mass-reproduction technique based upon a Gordon-type argument, and on the Jitomirskaya-Last extension of the Gilbert-Pearson theory of subordinacy.
... We would like to add that recently a number of remarkable properties of quasiperiodic operators with analytic potentials have been established based on the positivity of the Lyapunov exponents [12,4,14,10]. Theorem 1, therefore, adds to the collection of general corollaries of positive Lyapunov exponents for analytic potentials. ...
We study discrete Schroedinger operators on , where f(x) is a real analytic periodic function of period 1. We prove a general theorem relating the measure of the spectrum of to the measures of the spectra of its canonical rational approximants under the condition that the Lyapunov exponents of are positive. For the almost Mathieu operator () it follows that the measure of the spectrum is equal to for all real , , and all irrational .
This note gives a classification for the spectral measures of CMV matrices, which are special unitary five-diagonal matrices related to orthogonal polynomials on the unit circle, with respect to packing measures.
It is rigorously proven that the spectrum of the tight-binding Fibonacci Hamiltonian,H
mn=
m, n+1+
m, n–1+
m, n
[(n+1)]–[n]) where =(5–1)/2 and [] means integer part, is a Cantor set of zero Lebesgue measure for all real nonzero, and the spectral measures are purely singular continuous. This follows from a recent result by Kotani, coupled with the vanishing of the Lyapunov exponent in the spectrum.
1. The following objects are contemplated in this paper:— 1st. The demonstration of a fundamental theorem for the summation of integrals whose limits are determined by the roots of an algebraic equation. 2ndly. The application of that theorem to the problem of the comparison of algebraic transcendents. The immediate object of this application will in each case be the finite expression of the value of the sum of a series of integrals, Ʃ∫X dx , the differential coefficient, X, being an algebraic function, and the values of x at the limits being determined by the roots of an algebraic equation.
Synopsis
The theory of subordinacy is extended to all one-dimensional Schrödinger operatorsfor which the corresponding differential expression L = – d ² /(dr ² ) + V ( r ) is in the limit point case at both ends of an interval ( a , b ), with V ( r ) locally integrable. This enables a detailed classification of the absolutely continuous and singular spectra to be established in terms of the relative asymptotic behaviour of solutions of Lu = xu, x εℝ , as r → a and r → b . The result provides a rigorous but straightforward method of direct spectral analysis which has very general application, and somefurther properties of the spectrum are deduced from the underlying theory.
Many one-dimensional quasiperiodic systems based on the Fibonacci rule, such as the tight-binding HamiltonianH(Ö5 -1)/2(\sqrt 5 --1)/2
, give rise to the same recursion relation for the transfer matrices. It is proved that the wave functions and the norm of transfer matrices are polynomially bounded (critical regime) if and only if the energy is in the spectrum of the Hamiltonian. This solves a conjecture of Kohmoto and Sutherland on the power-law growth of the resistance in a one-dimensional quasicrystal.