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Jake Fillman, Darren Ong, Zhenghe Zhang
Spectral characteristics of the unitary critical almostMathieu operator
Communications in Mathematical Physics 351 (2017), 525–561.
Abstract:
We discuss spectral characteristics of a onedimensional quantum walk whose coins are distributed quasiperiodically. The unitary update rule of this quantum walk shares many spectral characteristics with the critical AlmostMathieu Operator; however, it possesses a feature not present in the AlmostMathieu Operator, namely singularity of the associated cocycles (this feature is, however, present in the socalled Extended Harper's Model). We show that this operator has empty absolutely continuous spectrum and that the Lyapunov exponent vanishes on the spectrum; hence, this model exhibits Cantor spectrum of zero Lebesgue measure for all irrational frequencies and arbitrary phase, which in physics is known as Hofstadter's butterfly. In fact, we will show something stronger, namely, that all spectral parameters in the spectrum are of critical type, in the language of Avila's global theory of analytic quasiperiodic cocycles. We further prove that it has empty point spectrum for each irrational frequency and away from a frequencydependent set of phases having Lebesgue measure zero. The key ingredients in our proofs are an adaptation of Avila's Global Theory to the present setting, selfduality via the Fourier transform, and a Johnsontype theorem for singular dynamically defined CMV matrices which characterizes their spectra as the set of spectral parameters at which the associated cocycles fail to admit a dominated splitting.


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Jake Fillman
Ballistic transport for limitperiodic Jacobi matrices with applications to quantum manybody problems
Communications in Mathematical Physics 350 (2017), 1275–1297.
Abstract:
We study Jacobi matrices that are uniformly approximated by periodic operators. We show that if the rate of approximation is sufficiently rapid, then the associated quantum dynamics are ballistic in a rather strong sense; namely, the (normalized) Heisenberg evolution of the position operator converges strongly to a selfadjoint operator that is injective on the space of absolutely summable sequences. In particular, this means that all transport exponents corresponding to welllocalized initial states are equal to one. Our result may be applied to a class of quantum manybody problems. Specifically, we establish a lower bound on the LiebRobinson velocity for an isotropic XY spin chain on the integers with limitperiodic couplings.


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Jake Fillman, Milivoje Lukic
Spectral homogeneity of limitperiodic Schrödinger operators
Journal of Spectral Theory 7 (2017), 387–406.
Abstract:
We prove that the spectrum of a limitperiodic Schrödinger operator is homogeneous in the sense of Carleson whenever the potential obeys the Pastur–Tkachenko condition. This implies that a dense set of limitperiodic Schrödinger operators have purely absolutely continuous spectrum supported on a homogeneous Cantor set. When combined with work of Gesztesy–Yuditskii, this also implies that the spectrum of a Pastur–Tkachenko potential has infinite gap length whenever the potential fails to be uniformly almost periodic.


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Jake Fillman, Darren Ong
Purely singular continuous spectrum for limitperiodic CMV operators with applications to quantum walks
Journal of Functional Analysis 272 (2017), 5107–5143.
Abstract:
We show that a generic element of a space of limitperiodic CMV operators has zeromeasure Cantor spectrum. We also prove a Craig–Simon type theorem for the density of states measure associated with a stochastic family of CMV matrices and use our construction from the first part to prove that the Craig–Simon result is optimal in general. We discuss applications of these results to a quantum walk model where the coins are arranged according to a limitperiodic sequence. The key ingredient in these results is a new formula which may be viewed as a relationship between the density of states measure of a CMV matrix and its Schur function.


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Jake Fillman
Purely singular continuous spectrum for Sturmian CMV matrices via strengthened Gordon Lemmas
Proceedings of the American Mathematical Society 145 (2017), 225–239.
Abstract:
The Gordon Lemma refers to a class of results in spectral theory which prove that strong local repetitions in the structure of an operator preclude the existence of eigenvalues for said operator. We expand on recent work of Ong and prove versions of the Gordon Lemma which are valid for CMV matrices and which do not restrict the parity of scales upon which repetitions occur. The key ingredient in our approach is a formula of Damanik–Fillman–Lukic–Yessen which relates two classes of transfer matrices for a given CMV operator. There are many examples to which our result can be applied. We apply our theorem to complete the classification of the spectral type of CMV matrices with Sturmian Verblunsky coefficients; we prove that such CMV matrices have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure for all (irrational) frequencies and all phases. We also discuss applications to CMV matrices with Verblunsky coefficients generated by general codings of rotations.


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Jake Fillman
Spectral homogeneity of discrete onedimensional limitperiodic operators
Journal of Spectral Theory 7 (2017), 201–226.
Abstract:
We prove that a dense subset of limit periodic operators have spectra which are homogeneous Cantor sets in the sense of Carleson. Moreover, by using work of Egorova, our examples have purely absolutely continuous spectrum. The construction is robust enough to extend the results to arbitrary $p$adic hulls by using the dynamical formalism proposed by Avila. The approach uses Floquet theory to break up the spectra of periodic approximants in a carefully controlled manner to produce Cantor spectrum and to establish the lower bounds needed to prove homogeneity.


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David Damanik, Jake Fillman, and Darren Ong
Spreading estimates for quantum walks on the integer lattice via powerlaw bounds on transfer matrices
Journal de Mathématiques Pures et Appliquées 105 (2016), 293–341.
Abstract:
We discuss spreading estimates for dynamical systems given by the iteration of an extended CMV matrix. Using a connection due to Cantero–Grünbaum–Moral–Velázquez, this enables us to study spreading rates for quantum walks in one spatial dimension. We prove several general results which establish quantitative upper and lower bounds on the spreading of a quantum walk in terms of estimates on a pair of associated matrix cocycles. To demonstrate the power and utility of these methods, we apply them to several concrete cases of interest. In the case where the coins are distributed according to an element of the Fibonacci subshift, we are able to rather completely describe the dynamics in a particular asymptotic regime. As a pleasant consequence, this supplies the first concrete example of a quantum walk with anomalous transport, to the best of our knowledge. We also prove ballistic transport for a quantum walk whose coins are periodically distributed.


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David Damanik, Jon Erickson, Jake Fillman, Gerhardt Hinkle, and Alan Vu
Quantum intermittency for sparse CMV matrices with an application to quantum walks on the halfline
Journal of Approximation Theory 208 (2016), 59–84.
Abstract:
We study the dynamics given by the iteration of a (halfline) CMV matrix with sparse, high barriers. Using an approach of Tcheremchantsev, we are able to explicitly compute the transport exponents for this model in terms of the given parameters. In light of the connection between CMV matrices and quantum walks on the halfline due to Cantero–Grünbaum–Moral–Velázquez, our result also allows us to compute transport exponents corresponding to a quantum walk which is sparsely populated with strong reflectors. To the best of our knowledge, this provides the first rigorous example of a quantum walk that exhibits quantum intermittency, i.e., nonconstancy of the transport exponents. When combined with the CMV version of the JitomirskayaLast theory of subordinacy and the general discretetime dynamical bounds from DamanikFillmanVance, we are able to exactly compute the Hausdorff dimension of the associated spectral measure.


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Jake Fillman, Yuki Takahashi, William Yessen
Mixed spectral regimes for square Fibonacci Hamiltonians
Journal of Fractal Geometry, 3 (2016), 377–405.
Abstract:
For the square tridiagonal Fibonacci Hamiltonian, we prove existence of an open set of parameters which yield mixed intervalCantor spectra (i.e. spectra containing an interval as well as a Cantor set), as well as mixed density of states measure (i.e. one whose absolutely continuous and singular continuous components are both nonzero). Using the methods developed in this paper, we also show existence of parameter regimes for the square continuum Fibonacci Schrodinger operator yielding mixed intervalCantor spectra. These examples provide the first explicit examples of an interesting phenomenon that has not hitherto been observed in aperiodic Hamiltonians. Moreover, while we focus only on the Fibonacci model, our techniques are equally applicable to models based on any twoletter primitive invertible substitution.


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David Damanik, Jake Fillman, Milivoje Lukic, and William Yessen
Characterizations of uniform hyperbolicity and spectra of CMV matrices
Discrete and Continuous Dynamical Systems – Series S 9 (2016), 1009–1023.
Abstract:
We provide an elementary proof of the equivalence of various notions of uniform hyperbolicity for a class of $\mathbb{GL}(2,\mathbb{C})$ cocycles and establish a Johnsontype 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.


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David Damanik, Jake Fillman, Milivoje Lukic, and William Yessen
Uniform hyperbolicity for Szegő cocycles and applications to random CMV matrices and the Ising model
International Mathematics Research Notices
2015 (2015), 7110–7129.
Abstract:
We consider products of the matrices associated with the Szegő recursion from the theory of orthogonal polynomials on the unit circle and show that under suitable assumptions, their norms grow exponentially in the number of factors. In the language of dynamical systems, this result expresses a uniform hyperbolicity statement. We present two applications of this result. On the one hand, we identify explicitly the almost sure spectrum of extended CMV matrices with nonnegative random Verblunsky coefficients. On the other hand, we show that no Ising model in one dimension exhibits a phase transition. Also, in the case of dynamically generated interaction couplings, we describe a gap labeling theorem for the Lee–Yang zeros in the thermodynamic limit.


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Charles Puelz, Mark Embree, and Jake Fillman
Spectral approximation for quasiperiodic Jacobi operators
Integral Equations and Operator Theory 82 (2015), 533–554.
Abstract:
Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals and in more general studies of structures exhibiting aperiodic order. The spectra of these selfadjoint operators can be quite exotic, such as Cantor sets, and their fine properties yield insight into associated dynamical systems. Quasiperiodic operators can be approximated by periodic ones, the spectra of which can be computed via two finite dimensional eigenvalue problems. Since long periods are necessary to get detailed approximations, both computational efficiency and numerical accuracy become a concern. We describe a simple method for numerically computing the spectrum of a period$K$ Jacobi operator in $O(K^2)$ operations, and use it to investigate the spectra of Schrödinger operators with Fibonacci, period doubling, and Thue–Morse potentials.


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David Damanik, Jake Fillman, and Robert Vance
Dynamics of unitary operators
Journal of Fractal Geometry 1 (2014), 391–425.
Abstract:
We consider the iteration of a unitary operator on a separable Hilbert space and study the spreading rates of the associated discretetime dynamical system relative to a given orthonormal basis. We prove lower bounds for the transport exponents, which measure the timeaveraged spreading on a powerlaw 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 continuumtime dynamical system associated with a selfadjoint 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 timehomogeneous quantum walk on the integers. For the particular case of the timehomogeneous Fibonacci quantum walk, we illustrate how these components work together and produce explicit lower bounds for the transport exponents associated with this model.


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David Damanik, Jake Fillman, and Anton Gorodetski
Continuum Schrödinger operators associated with aperiodic subshifts
Annales Henri Poincaré , 15 (2014), 1123–1144.
*This paper was awarded the 2014 Annales Henri Poincaré Prize
Abstract:
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 FrickeVogt 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 socalled pseudo bands, which have been pointed out in the physics literature.
