Almost localization and almost reducibility

By Artur Avila, S. Jitomirskaya. 2008

210 citations recorded by the source corpus.

We develop a quantitative version of Aubry duality and use it to obtain several sharp estimates for the dynamics of Schr\"odinger cocycles associated to a non-perturbatively small analytic potential and Diophantine frequency. In particular, we establish the full version of Eliasson's reducibility theory in this regime (our approach actually leads to improvements even in the perturbative regime: we are able to show, for all energies, ``almost reducibility'' in some band of analyticity). We also prove 1/2-H\"older continuity of the integrated density of states. For the almost Mathieu operator, our results hold through the entire regime of sub-critical coupling and imply also the dry version of the Ten Martini Problem for the concerned parameters.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e1748c03fc1288a85407353a29c5d1e192badc53

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In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

Analytic Quasi-Perodic Cocycles with Singularities and the Lyapunov Exponent of Extended Harper’s Model (2010, background): “)Analysis of the Lyapunov exponent serves as an important ingredient in spectral analysis. For almost periodic Schrodinger operators with analytic potentials the almost reducibility conjecture (ARC) [2, 3, 5, 1] claims that sub-critical behavior of the Schrodinger cocycle on the spectrum implies pur”
Sharp phase transitions for the almost Mathieu operator (2015, background): “Recently, Avila-Jitomirskaya [10] showed that if α is Diophantine, then Hλ,α,θ is purely absolutely continuous for every θ ∈ R.”
EQUILIBRIUM MEASURES AND CAPACITIES IN SPECTRAL THEORY (2007, background): “ hand, Last [74] proved that for |λ| <2 and all irrational αthat the spectrum is a.c. for almost all θ(now known for all θ[71, 75]). It is now known the spectrum in this region is purely a.c. (see [9, 8, 6]). At the special point λ= 2, it is known that for all irrational α, the spectrum has measu”
Embedding of Analytic Quasi-Periodic Cocycles into Analytic Quasi-Periodic Linear Systems and its Applications (2012, methodology): “Avila and Jitomirskaya [7] further developed a quantitative Aubry duality to prove that almost localization implies almost reducibility for the dual model.”
Rigidity results for quasiperiodic SL(2, R) -cocycles (2010, methodology): “In the analytic case this theorem is also true, by dierent techniques, as a consequence of [ 3 ], [2], [5] where, in addition, the Diophantine condition on can be relaxed to be limn!1 logq”
Almost reducibility and absolute continuity I (2010, background): “allowed to diverge). Essentially by deﬁnition, the concept of almost reducibility “captures” the domain of applicability of “local theories” of cocycles close to constant. The basic hope expressed by [AJ1] can be thus expressed in the form of the Almost Reducibility Conjecture (ARC): subcriticality ”
Schrödinger operators with dynamically defined potentials (2014, background): “Going beyond that, Avila and Jitomirskaya developed a quantitative formulation of Aubry duality in [17] and used it to prove the following stronger result.”
A KAM scheme for SL(2,R) cocycles with Liouvillean frequencies (2010, methodology): “s [AK3]. For this reason, it is unlikely that a positive solution to the problem could ever be achieved by a KAM method (since KAM methods tend to generalize to weaker regularity). On the other hand, [AJ2] showed that the closeness quantiﬁer in Eliasson’s Theorem was α-independent, as long as αwas D”