The Hölder continuity of spectral measures of an extended CMV matrix

Munger, Paul; Ong, Darren C.

doi:10.1063/1.4895762

Cited by 11 publications

(14 citation statements)

References 17 publications

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“…The following result was shown in [17]. It is an adaptation of a result shown by Damanik, Killip, and Lenz in the Schrödinger context [4].…”

Section: Spectral Regularity Via Subordinacy Theorymentioning

confidence: 51%

“…This theorem was stated and proved in [5]. The proof given there used Proposition 4.4, proved in [17], and Proposition 3.9 and Corollaries 3.11 and 3.13, proved in this paper. Thus, these three papers work together in establishing Theorem 4.5.…”

mentioning

confidence: 68%

“…The Fibonacci Quantum Walk. We discuss the special case of the Fibonacci quantum walk, which is an example that requires the full extent of the machinery developed in this paper in conjunction with the subordinacy result from [17] described in Subsection 4.1. In this example the sequence of coins takes only two different values, and the order in which these two unitary 2 × 2 matrices occur is determined by an element of the Fibonacci subshift.…”

Section: 3mentioning

confidence: 99%

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Dynamics of unitary operators

Damanik¹,

Fillman²,

Vance³

2015

J. Fractal Geom.

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Abstract. We consider the iteration of a unitary operator on a separable Hilbert space and study the spreading rates of the associated discrete-time dynamical system relative to a given orthonormal basis. We prove lower bounds for the transport exponents, which measure the time-averaged spreading on a power-law scale, in terms of dimensional properties of the spectral measure associated with the unitary operator and the initial state. These results are the unitary analog of results established in recent years for the dynamics of the Schrödinger equation, which is a continuum-time dynamical system associated with a self-adjoint operator. We discuss how these general results may be studied by means of subordinacy theory in cases where the unitary operator is given by a CMV matrix. An example of particular interest in which this scenario arises is given by a time-homogeneous quantum walk on the integers. For the particular case of the time-homogeneous Fibonacci quantum walk, we illustrate how these components work together and produce explicit lower bounds for the transport exponents associated with this model.Mathematics Subject Classification (2010). 42C05; 82B41.

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“…The following result was shown in [17]. It is an adaptation of a result shown by Damanik, Killip, and Lenz in the Schrödinger context [4].…”

Section: Spectral Regularity Via Subordinacy Theorymentioning

confidence: 51%

mentioning

confidence: 68%

Section: 3mentioning

confidence: 99%

See 1 more Smart Citation

Dynamics of unitary operators

Damanik¹,

Fillman²,

Vance³

2015

J. Fractal Geom.

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“…These estimates will be proved in the present subsection. The other steps have been addressed (in a general context) in separate publications [10,25] and the relevant results from those papers are summarized in the appendix. Define T to be the transfer matrix corresponding to a Verblunsky coefficient of zero, and A, B the transfer matrices corresponding to θ a and θ b .…”

Section: Quantum Walk In a Fibonacci Environmentmentioning

confidence: 99%

“…for L ≥ 1, uniformly in ω. As before, the constant C 1 (z) is irrelevant to the long-term spreading behavior, so it is not calculated here, and the treatment in [25] contains an expression for it.…”

Section: Quantum Walk In a Fibonacci Environmentmentioning

confidence: 99%

Orthogonal Polynomials on the Unit Circle with Fibonacci Verblunsky Coefficients, II. Applications

2013

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Abstract. We consider CMV matrices with Verblunsky coefficients determined in an appropriate way by the Fibonacci sequence and present two applications of the spectral theory of such matrices to problems in mathematical physics. In our first application we estimate the spreading rates of quantum walks on the line with time-independent coins following the Fibonacci sequence. The estimates we obtain are explicit in terms of the parameters of the system. In our second application, we establish a connection between the classical nearest neighbor Ising model on the one-dimensional lattice in the complex magnetic field regime, and CMV operators. In particular, given a sequence of nearest-neighbor interaction couplings, we construct a sequence of Verblunsky coefficients, such that the support of the Lee-Yang zeros of the partition function for the Ising model in the thermodynamic limit coincides with the essential spectrum of the CMV matrix with the constructed Verblunsky coefficients. Under certain technical conditions, we also show that the zeros distribution measure coincides with the density of states measure for the CMV matrix.

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Continuum Schrödinger Operators Associated With Aperiodic Subshifts

Damanik

Fillman

Gorodetski

2013

Ann. Henri Poincaré

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We study Schrödinger operators on the real line whose potentials are generated by an underlying ergodic subshift over a finite alphabet and a rule that replaces symbols by compactly supported potential pieces. We first develop the standard theory that shows that the spectrum and the spectral type are almost surely constant, and that identifies the almost sure absolutely continuous spectrum with the essential closure of the set of energies with vanishing Lyapunov exponent. Using results of Damanik-Lenz and Klassert-Lenz-Stollmann, we also show that the spectrum is a Cantor set of zero Lebesgue measure if the subhift satisfies the Boshernitzan condition and the potentials are aperiodic and irreducible. We then study the case of the Fibonacci subshift in detail and prove results for the local Hausdorff dimension of the spectrum at a given energy in terms of the value of the associated Fricke-Vogt invariant. These results are elucidated for some simple choices of the local potential pieces, such as piecewise constant ones and local point interactions. In the latter special case, our results explain the occurrence of so-called pseudo bands, which have been pointed out in the physics literature.

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The Hölder continuity of spectral measures of an extended CMV matrix

Cited by 11 publications

References 17 publications

Dynamics of unitary operators

Dynamics of unitary operators

Orthogonal Polynomials on the Unit Circle with Fibonacci Verblunsky Coefficients, II. Applications

Continuum Schrödinger Operators Associated With Aperiodic Subshifts

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