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51
Spectra of selfadjoint extensions and applications to solvable Schrödinger operators
, 2007
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Spectral and quantum dynamical properties of the weakly coupled Fibonacci Hamiltonian
, 2010
"... We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We prov ..."
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Cited by 23 (10 self)
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We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We prove that the thickness tends to infinity and, consequently, the Hausdorff dimension of the spectrum tends to one. We also show that at small coupling, all gaps allowed by the gap labeling theorem are open and the length of every gap tends to zero linearly. Moreover, for sufficiently small coupling, the sum of the spectrum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by EvenDar Mandel and Lifshitz. Finally, we provide explicit upper and lower bounds for the solutions to the difference equation and use them to
Recent developments in quantum mechanics with magnetic fields
 Proc. of Symposia in Pure Math. Vol 76 Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday Part
, 2006
"... We present a review on the recent developments concerning rigorous mathematical results on Schrödinger operators with magnetic fields. This paper is dedicated to the sixtieth birthday of Barry Simon. ..."
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Cited by 17 (0 self)
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We present a review on the recent developments concerning rigorous mathematical results on Schrödinger operators with magnetic fields. This paper is dedicated to the sixtieth birthday of Barry Simon.
Cantor and band spectra for periodic quantum graphs with magnetic fields
 Comm. Math. Phys
"... ABSTRACT. We provide an exhaustive spectral analysis of the twodimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lya ..."
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Cited by 15 (3 self)
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ABSTRACT. We provide an exhaustive spectral analysis of the twodimensional periodic square graph lattice with a magnetic field. We show that the spectrum consists of the Dirichlet eigenvalues of the edges and of the preimage of the spectrum of a certain discrete operator under the discriminant (Lyapunov function) of a suitable KronigPenney Hamiltonian. In particular, between any two Dirichlet eigenvalues the spectrum is a Cantor set for an irrational flux, and is absolutely continuous and has a band structure for a rational flux. The Dirichlet eigenvalues can be isolated or embedded, subject to the choice of parameters. Conditions for both possibilities are given. We show that generically there are infinitely many gaps in the spectrum, and the BetheSommerfeld conjecture fails in this case.
SPECTRAL PROPERTIES OF SCHRÖDINGER OPERATORS ARISING IN THE STUDY OF QUASICRYSTALS
, 2012
"... We survey results that have been obtained for selfadjoint 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 onedimensional case, and in particula ..."
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Cited by 14 (7 self)
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We survey results that have been obtained for selfadjoint 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 onedimensional 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
Absolute continuity of the integrated density of states for the almost Mathieu operator with noncritical coupling
, 2007
"... We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is noncritical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measuretheoretical case of P ..."
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Cited by 14 (7 self)
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We show that the integrated density of states of the almost Mathieu operator is absolutely continuous if and only if the coupling is noncritical. We deduce for subcritical coupling that the spectrum is purely absolutely continuous for almost every phase, settling the measuretheoretical case of Problem 6 of Barry Simon’s list of Schrödinger operator problems for the twentyfirst century.
A Nonperturbative Eliasson’s Reducibility Theorem
"... This paper is concerned with discrete, onedimensional Schrödinger operators with real analytic potentials and one Diophantine frequency. Using localization and duality we show that almost every point in the spectrum admits a quasiperiodic Bloch wave if the potential is smaller than a certain const ..."
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Cited by 11 (1 self)
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This paper is concerned with discrete, onedimensional Schrödinger operators with real analytic potentials and one Diophantine frequency. Using localization and duality we show that almost every point in the spectrum admits a quasiperiodic Bloch wave if the potential is smaller than a certain constant which does not depend on the precise Diophantine conditions. The associated firstorder system, a quasiperiodic skewproduct, is shown to be reducible for almost all values of the energy. This is a partial nonperturbative generalization of a reducibility theorem by Eliasson. We also extend nonperturbatively the genericity of Cantor spectrum for these Schrödinger operators. Finally we prove that in our setting, Cantor spectrum implies the existence of a Gδset of energies whose Schrödinger cocycle is not reducible to constant coefficients. Keywords: Quasiperiodic Schrödinger operators, Harperlike equations, reducibility, Floquet theory, quasiperiodic cocycles, skewproduct,
The creation of strange nonchaotic attractors in nonsmooth saddlenode bifurcations
, 2008
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Examples of discontinuity of Lyapunov Exponent in smooth quasiperiodic cocycles, preprint
"... We study the regularity of the Lyapunov exponent for quasiperiodic cocycles (Tω, A) where Tω is an irrational rotation x → x + 2piω on S1 and A ∈ Cl(S1, SL(2,R)), 0 ≤ l ≤ ∞. For any fixed l = 0, 1, 2, · · ·, ∞ and any fixed ω of boundedtype, we construct Dl ∈ Cl(S1, SL(2,R)) such that the Lyapu ..."
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Cited by 8 (2 self)
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We study the regularity of the Lyapunov exponent for quasiperiodic cocycles (Tω, A) where Tω is an irrational rotation x → x + 2piω on S1 and A ∈ Cl(S1, SL(2,R)), 0 ≤ l ≤ ∞. For any fixed l = 0, 1, 2, · · ·, ∞ and any fixed ω of boundedtype, we construct Dl ∈ Cl(S1, SL(2,R)) such that the Lyapunov exponent is not continuous at Dl in Cltopology. We also construct such examples in a smaller Schrödinger class. 1