Resistance of one-dimensional quasicrystals

Iochum, Bruno; Raymond, L.; Testard, D.

doi:10.1016/0378-4371(92)90426-q

Cited by 42 publications

(38 citation statements)

References 40 publications

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“…Interestingly enough, it was sufficient also in that case to study traces. This less trivial transition from traces to norms can be achieved in the situation at hand using an observation of Iochum and Testard [47]; see also [46].…”

Section: The Fibonacci Operatormentioning

confidence: 99%

Upper bounds in quantum dynamics

Damanik

Tcheremchantsev²

2006

J. Amer. Math. Soc.

138

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We develop a general method to bound the spreading of an entire wavepacket under Schrödinger dynamics from above. This method derives upper bounds on time-averaged 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 quasi-periodic 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 well-known 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.

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Section: The Fibonacci Operatormentioning

confidence: 99%

Upper bounds in quantum dynamics

Damanik

Tcheremchantsev²

2006

J. Amer. Math. Soc.

138

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“…Nous donnons ici une borne inférieure pour la dimension de boîte du spectre de l'opérateur et la borne dynamique inférieure qui en découle (voir [5,11]). Théorème 0.6.…”

Section: Sci Paris Ser I 346 (2008)unclassified

Dynamical bounds for Sturmian Schrödinger operators

Marin

2008

Comptes Rendus. Mathématique

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“…Notice that this improves on the statement that the Lyapunov exponent vanishes on the spectrum. An extension to Sturmian subshifts whose slope has (essentially) bounded continued fraction coefficients was obtained by Iochum et al [79]. Note that upper bounds for transfer matrix norms yield the input to Theorem 7.4 and one half of the input to Theorem 7.2.…”

Section: Quantum Dynamicsmentioning

confidence: 99%

Strictly ergodic subshifts and associated operators

Damanik¹

2007

Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday

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Dedicated to Barry Simon on the occasion of his 60th birthday.Abstract. 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.

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Resistance of one-dimensional quasicrystals

Cited by 42 publications

References 40 publications

Upper bounds in quantum dynamics

Upper bounds in quantum dynamics

Dynamical bounds for Sturmian Schrödinger operators

Strictly ergodic subshifts and associated operators

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