Browsing by Author "Gorodetski, Anton"
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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 Spectral Properties of Schrödinger Operators Arising in the Study of Quasicrystals(2012-10) Damanik, David; Embree, Mark; Gorodetski, AntonWe survey results that have been obtained for self-adjoint operators, and especially Schrödinger operators, associated with mathematical models of quasicrystals. After presenting general results that hold in arbitrary dimensions, we focus our attention on the one-dimensional case, and in particular on several key examples. The most prominent of these is the Fibonacci Hamiltonian, for which much is known by now and to which an entire section is devoted here. Other examples that are discussed in detail are given by the more general class of Schrödinger operators with Sturmian potentials. We put some emphasis on the methods that have been introduced quite recently in the study of these operators, many of them coming from hyperbolic dynamics. We conclude with a multitude of numerical calculations that illustrate the validity of the known rigorous results and suggest conjectures for further exploration.Item The Density of States Measure of the Weakly Coupled Fibonacci Hamiltonian(Springer, 2012) Damanik, David; Gorodetski, AntonWe consider the density of states measure of the Fibonacci Hamiltonian and show that, for small values of the coupling constant V , this measure is exact-dimensional and the almost everywhere value dV of the local scaling exponent is a smooth function of V , is strictly smaller than the Hausdor dimension of the spectrum, and converges to one as V tends to zero. The proof relies on a new connection between the density of states measure of the Fibonacci Hamiltonian and the measure of maximal entropy for the Fibonacci trace map on the non-wandering set in the V -dependent invariant surface. This allows us to make a connection between the spectral problem at hand and the dimension theory of dynamical systems.