Results 1  10
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27
Uniform Spectral Properties Of OneDimensional Quasicrystals, IV. QuasiSturmian Potentials
 I. Absence of eigenvalues, Commun. Math. Phys
, 2000
"... We consider discrete onedimensional Schrodinger operators with quasiSturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, ..."
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Cited by 79 (44 self)
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We consider discrete onedimensional Schrodinger operators with quasiSturmian potentials. We present a new approach to the trace map dynamical system which is independent of the initial conditions and establish a characterization of the spectrum in terms of bounded trace map orbits. Using this, it is shown that the operators have purely singular continuous spectrum and their spectrum is a Cantor set of Lebesgue measure zero. We also exhibit a subclass having purely ffcontinuous spectrum. All these results hold uniformly on the hull generated by a given potential.
Duality and singular continuous spectrum in the almost Mathieu equation
, 1997
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Fine structure of the zeros of orthogonal polynomials, II. OPUC with competing exponential decay
, 2004
"... We present a complete theory of the asymptotics of the zeros of OPUC with Verblunsky coefficients αn = �L ℓ=1 Cℓbn ℓ + O((b∆) n) where ∆ < 1 and bℓ  = b < 1. ..."
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Cited by 36 (13 self)
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We present a complete theory of the asymptotics of the zeros of OPUC with Verblunsky coefficients αn = �L ℓ=1 Cℓbn ℓ + O((b∆) n) where ∆ < 1 and bℓ  = b < 1.
Anderson Localization for the Almost Mathieu Equation: A Nonperturbative Proof.
"... We prove that for any diophantine rotation angle ! and a.e. phase ` the almost Mathieu operator (H(`)\Psi) n = \Psi n\Gamma1 + \Psi n+1 + cos(2ß(` + n!))\Psi n has pure point spectrum with exponentially decaying eigenfunctions for 15: We also prove the existence of some pure point spectrum for an ..."
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Cited by 22 (9 self)
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We prove that for any diophantine rotation angle ! and a.e. phase ` the almost Mathieu operator (H(`)\Psi) n = \Psi n\Gamma1 + \Psi n+1 + cos(2ß(` + n!))\Psi n has pure point spectrum with exponentially decaying eigenfunctions for 15: We also prove the existence of some pure point spectrum for any 5:4: Permanent address: International Institute of Earthquake Prediction Theory and Mathematical Geophysics. Moscow, Russia . 1. INTRODUCTION In this paper we study localization for the almostMathieu operator on ` 2 (Z) : (H(`)\Psi) n = \Psi n\Gamma1 + \Psi n+1 + cos(2ß(` + n!))\Psi n The almostMathieu operator attracted a lot of interest especially in the last decade. For references before 1985 see [1]. Some of the later references are [27 ]. While it is very well understood and commonly believed that for diophantine ! and jj ? 2 the operator H(`) should have pure point spectrum with exponentially decaying eigenfunctions for almost every ` this is not yet rigorously proved ....
Twisted index theory on good orbifolds, I: noncommutative Bloch theory
 Commun. Contemp. Math. Vol1
, 1999
"... Abstract. We study the twisted index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group. We apply these results to obtain qualitative results on real and complex hyperbolic spaces in 2 an ..."
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Cited by 11 (7 self)
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Abstract. We study the twisted index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group. We apply these results to obtain qualitative results on real and complex hyperbolic spaces in 2 and 4 dimensions, related to generalizations of the BetheSommerfeld conjecture and the Ten Martini Problem, on the spectrum of self adjoint elliptic operators which are invariant under a projective action of a discrete cocompact group.
Spectral estimates for periodic Jacobi matrices
, 2008
"... We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on ℓ 2 (Z) of the form (Hψ)n = an−1ψn−1 + bnψn + anψn+1, where an = an+q and bn = bn+q are periodic sequences of real numbers. The results are based on a study of the quasimomentum k(z) corresponding t ..."
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Cited by 9 (4 self)
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We obtain bounds for the spectrum and for the total width of the spectral gaps for Jacobi matrices on ℓ 2 (Z) of the form (Hψ)n = an−1ψn−1 + bnψn + anψn+1, where an = an+q and bn = bn+q are periodic sequences of real numbers. The results are based on a study of the quasimomentum k(z) corresponding to H. We consider k(z) as a conformal mapping in the complex plane. We obtain the trace identities which connect integrals of the Lyapunov exponent over the gaps with the normalised traces of powers of H.
Twisted higher index theory on good orbifolds and fractional quantum numbers
"... Abstract. In this paper, we study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group, and we apply these results to obtain qualitative results, related to general ..."
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Cited by 6 (1 self)
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Abstract. In this paper, we study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group, and we apply these results to obtain qualitative results, related to generalizations of the BetheSommerfeld conjecture, on the spectrum of self adjoint elliptic operators which are invariant under a projective action of the orbifold fundamental group. We also compute the range of the higher traces on Ktheory, which we then apply to compute the range of values of the Hall conductance in the quantum Hall effect on the hyperbolic plane. The new phenomenon that we observe in this case is that the Hall conductance again has plateaus at all energy levels belonging to any gap in the spectrum of the Hamiltonian, where it is now shown to be equal to an integral multiple of a fractional valued invariant. Moreover the set of possible denominators is finite and has been explicitly determined. It is plausible that this might shed light on the mathematical mechanism responsible for fractional quantum numbers.
Existence Of NonUniform Cocycles On Uniquely Ergodic Systems
"... We study existence of nonuniform continuous SL(2; R)valued cocycles over uniquely ergodic dynamical systems. We present a class of subshifts over nite alphabets on which every locally constant cocycle is uniform. On the other hand, we also show that every irrational rotation admits nonuniform coc ..."
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Cited by 5 (4 self)
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We study existence of nonuniform continuous SL(2; R)valued cocycles over uniquely ergodic dynamical systems. We present a class of subshifts over nite alphabets on which every locally constant cocycle is uniform. On the other hand, we also show that every irrational rotation admits nonuniform cocycles. Finally, we discuss a characterization of uniformity in terms of a hyperbolicity condition.
Discrete random electromagnetic Laplacians
 in the Mathematical Physics Preprint Archive, mp arc@math.utexas.edu
, 1995
"... We consider discrete random magnetic Laplacians in the plane and discrete random electromagnetic Laplacians in higher dimensions. The existence of these objects relies on a theorem of FeldmanMoore which was generalized by Lind to the nonabelian case. For example, it allows to realize ergodic Schrod ..."
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Cited by 5 (4 self)
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We consider discrete random magnetic Laplacians in the plane and discrete random electromagnetic Laplacians in higher dimensions. The existence of these objects relies on a theorem of FeldmanMoore which was generalized by Lind to the nonabelian case. For example, it allows to realize ergodic Schrodinger operators with stationary independent magnetic fields on discrete two dimensional lattices including also nonperiodic situations like Penrose lattices. The theorem is generalized here to higher dimensions. The Laplacians obtained from the electromagnetic vector potential are elements of a von Neumann algebra constructed from the underlying dynamical system respectively from the ergodic equivalence relation. They generalize Harper operators which correspond to constant magnetic fields. For independent identically distributed magnetic fields and special Anderson models, we compute the density of states using a random walk expansion. Mathematics subject classification: 28D15, 47A10, 47A3...
Ergodic Theory and Discrete OneDimensional Random Schrödinger Operators: Uniform Existence of the Lyapunov Exponent
"... We review recent results which relate spectral theory of discrete onedimensional Schrödinger operators over strictly ergodic systems to uniform existence of the Lyapunov exponent. In combination with suitable ergodic theorems this allows one to establish Cantor spectrum of Lebesgue measure zero ..."
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Cited by 4 (3 self)
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We review recent results which relate spectral theory of discrete onedimensional Schrödinger operators over strictly ergodic systems to uniform existence of the Lyapunov exponent. In combination with suitable ergodic theorems this allows one to establish Cantor spectrum of Lebesgue measure zero for a large class of quasicrystal Schrödinger operators. The results can also be used to study nonuniformity of cocycles. While most