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Strictly ergodic subshifts and associated operators
, 2005
"... We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have in ..."
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Cited by 13 (8 self)
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We consider ergodic families of Schrödinger operators over base dynamics given by strictly ergodic subshifts on finite alphabets. It is expected that the majority of these operators have purely singular continuous spectrum supported on a Cantor set of zero Lebesgue measure. These properties have indeed been established for large classes of operators of this type over the course of the last twenty years. We review the mechanisms leading to these results and briefly discuss analogues for CMV matrices.
Upper bounds in quantum dynamics
 J. Amer. Math. Soc
"... Abstract. We develop a general method to bound the spreading of an entire wavepacket under Schrödinger dynamics from above. This method derives upper bounds on timeaveraged moments of the position operator from lower bounds on norms of transfer matrices at complex energies. This general result is a ..."
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Cited by 8 (6 self)
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Abstract. We develop a general method to bound the spreading of an entire wavepacket under Schrödinger dynamics from above. This method derives upper bounds on timeaveraged moments of the position operator from lower bounds on norms of transfer matrices at complex energies. This general result is applied to the Fibonacci operator. We find that at sufficiently large coupling, all transport exponents take values strictly between zero and one. This is the first rigorous result on anomalous transport. For quasiperiodic potentials associated with trigonometric polynomials, we prove that all lower transport exponents and, under a weak assumption on the frequency, all upper transport exponents vanish for all phases if the Lyapunov exponent is uniformly bounded away from zero. By a wellknown result of Herman, this assumption always holds at sufficiently large coupling. For the particular case of the almost Mathieu operator, our result applies for coupling greater than two. 1.
A note on fractional moments for the onedimensional continuum Anderson model
 J. Math. Anal. Appl
"... Abstract. We give a proof of dynamical localization in the form of exponential decay of spatial correlations in the time evolution for the onedimensional continuum Anderson model via the fractional moments method. This follows via exponential decay of fractional moments of the Green function, which ..."
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Cited by 2 (1 self)
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Abstract. We give a proof of dynamical localization in the form of exponential decay of spatial correlations in the time evolution for the onedimensional continuum Anderson model via the fractional moments method. This follows via exponential decay of fractional moments of the Green function, which is shown to hold at arbitrary energy and for any singlesite distribution with bounded, compactly supported density. 1.
A MATRIXVALUED POINT INTERACTIONS MODEL
, 809
"... Abstract. We study a matrixvalued Schrödinger operator with random point interactions. We prove the absence of absolutely continuous spectrum for this operator by proving that away from a discrete set its Lyapunov exponents do not vanish. For this we use a criterion by Gol’dsheid and Margulis and w ..."
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Abstract. We study a matrixvalued Schrödinger operator with random point interactions. We prove the absence of absolutely continuous spectrum for this operator by proving that away from a discrete set its Lyapunov exponents do not vanish. For this we use a criterion by Gol’dsheid and Margulis and we prove the Zariski denseness, in the symplectic group, of the group generated by the transfer matrices. Then we prove estimates on the transfer matrices which lead to the Hölder continuity of the Lyapunov exponents. After proving the existence of the integrated density of states of the operator, we also prove its Hölder continuity by proving a Thouless formula which links the integrated density of states to the sum of the positive Lyapunov exponents.
QUANTUM HAMILTONIANS WITH QUASIBALLISTIC DYNAMICS AND POINT SPECTRUM
, 2007
"... Abstract. Consider the family of Schrödinger operators (and also its Dirac version) on ℓ 2 (Z) or ℓ 2 (N) H W ω,S = ∆ + λF(S n ω) + W, ω ∈ Ω, where S is a transformation on (compact metric) Ω, F a real Lipschitz function and W a (sufficiently fast) powerdecaying perturbation. Under certain conditi ..."
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Abstract. Consider the family of Schrödinger operators (and also its Dirac version) on ℓ 2 (Z) or ℓ 2 (N) H W ω,S = ∆ + λF(S n ω) + W, ω ∈ Ω, where S is a transformation on (compact metric) Ω, F a real Lipschitz function and W a (sufficiently fast) powerdecaying perturbation. Under certain conditions it is shown that H W ω,S presents quasiballistic dynamics for ω in a dense Gδ set. Applications include potentials generated by rotations of the torus with analytic condition on F, doubling map, Axiom A dynamical systems and the Anderson model. If W is a rank one perturbation, examples of H W ω,S with quasiballistic dynamics and point spectrum are also presented. 1.
SPECTRAL AND LOCALIZATION PROPERTIES FOR THE ONEDIMENSIONAL BERNOULLI DISCRETE DIRAC OPERATOR
, 2005
"... Abstract. A 1D Dirac tightbinding model is considered and it is shown that its nonrelativistic limit is the 1D discrete Schrödinger model. For random Bernoulli potentials taking two values (without correlations), for typical realizations and for all values of the mass, it is shown that its spectrum ..."
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Abstract. A 1D Dirac tightbinding model is considered and it is shown that its nonrelativistic limit is the 1D discrete Schrödinger model. For random Bernoulli potentials taking two values (without correlations), for typical realizations and for all values of the mass, it is shown that its spectrum is pure point, whereas the zero mass case presents dynamical delocalization for specific values of the energy. The massive case presents dynamical localization (excluding some particular values of the energy). Finally, for general potentials the dynamical moments for distinct masses are compared, especially the massless and massive Bernoulli cases. 1.
ftp ejde.math.txstate.edu (login: ftp) POSITIVITY OF LYAPUNOV EXPONENTS FOR ANDERSONTYPE MODELS ON TWO COUPLED STRINGS
"... Abstract. We study two models of Andersontype random operators on two deterministically coupled continuous strings. Each model is associated with independent, identically distributed fourbyfour symplectic transfer matrices, which describe the asymptotics of solutions. In each case we use a criter ..."
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Abstract. We study two models of Andersontype random operators on two deterministically coupled continuous strings. Each model is associated with independent, identically distributed fourbyfour symplectic transfer matrices, which describe the asymptotics of solutions. In each case we use a criterion by Gol’dsheid and Margulis (i.e. Zariski denseness of the group generated by the transfer matrices in the group of symplectic matrices) to prove positivity of both leading Lyapunov exponents for most energies. In each case this implies almost sure absence of absolutely continuous spectrum (at all energies in the first model and for sufficiently large energies in the second model). The methods used allow for singularly distributed random parameters, including Bernoulli distributions. 1.