Browsing by Author "Fillman, Jake"

Now showing 1 - 8 of 8
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    Ergodic Schrödinger operators in the infinite measure setting
    (EMS Press, 2021) Boshernitzan, Michael; Damanik, David; Fillman, Jake; Lukic, Milivoje
    We 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.
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    Johnson–Schwartzman gap labelling for ergodic Jacobi matrices
    (EMS Press, 2023) Damanik, David; Fillman, Jake; Zhang, Zhenghe
    We 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.
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    Limit-periodic Schrödinger operators with Lipschitz continuous IDS
    (American Mathematical Society, 2019) Damanik, David; Fillman, Jake
    We 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.
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    Multidimensional Almost-Periodic Schrödinger Operators with Cantor Spectrum
    (Springer, 2019) Damanik, David; Fillman, Jake; Gorodetski, Anton
    We 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.
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    Multidimensional Schrödinger operators whose spectrum features a half-line and a Cantor set
    (Elsevier, 2021) Damanik, David; Fillman, Jake; Gorodetski, Anton
    We 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.
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    Positive Lyapunov exponents and a Large Deviation Theorem for continuum Anderson models, briefly
    (Elsevier, 2019) Bucaj, Valmir; Damanik, David; Fillman, Jake; Gerbuz, Vitaly; VandenBoom, Tom; Wang, Fengpeng; Zhang, Zhenghe
    In this short note, we prove positivity of the Lyapunov exponent for 1D continuum Anderson models by leveraging some classical tools from inverse spectral theory. The argument is much simpler than the existing proof due to Damanik–Sims–Stolz, and it covers a wider variety of random models. Along the way we note that a Large Deviation Theorem holds uniformly on compacts.
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    Random Hamiltonians with arbitrary point interactions in one dimension
    (Elsevier, 2021) Damanik, David; Fillman, Jake; Helman, Mark; Kesten, Jacob; Sukhtaiev, Selim
    We consider disordered Hamiltonians given by the Laplace operator subject to arbitrary random self-adjoint singular perturbations supported on random discrete subsets of the real line. Under minimal assumptions on the type of disorder, we prove the following dichotomy: Either every realization of the random operator has purely absolutely continuous spectrum or spectral and exponential dynamical localization hold. In particular, we establish Anderson localization for Schrödinger operators with Bernoulli-type random singular potential and singular density.
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    Zero measure spectrum for multi-frequency Schrödinger operators
    (EMS Press, 2022) Chaika, Jon; Damanik, David; Fillman, Jake; Gohlke, Philipp
    Building on works of Berthé–Steiner–Thuswaldner and Fogg–Nous, we show that on the two-dimensional torus, Lebesgue almost every translation admits a natural coding such that the associated subshift satisfies the Boshernitzan criterion. As a consequence, we show that for these torus translations, every quasi-periodic potential can be approximated uniformly by one for which the associated Schrödinger operator has Cantor spectrum of zero Lebesgue measure. We also describe a framework that can allow this to be extended to higher-dimensional tori.