Holder Continuity of the Integrated Density of States for Quasi-Periodic Schrodinger Equations and Averages of Shifts of Subharmonic Functions

Goldstein, Michael; Schlag, Wilhelm

doi:10.2307/3062114

Cited by 191 publications

(293 citation statements)

References 26 publications

Supporting

Mentioning

280

Contrasting

Unclassified

Order By: Relevance

“…The following result improves on the uniform upper bound on the monodromy matrices from [BG00] and [GS01]. The (log N ) A error here (rather than N σ , say, as in [BG00] and [GS01]) is crucial for the study of the distribution of the zeros of the determinants and eigenvalues, see Proposition 4.3 in [GS08].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 62%

“…The key analytical tool for the study of periodic boundary conditions consists of a large deviation estimate for the trace of the propagator matrix M N which we derive in Section 3. In deriving these large deviation estimates we rely on techniques developed by Bourgain and the first author in [BG00], and by the authors in [GS01] and [GS07]. Loosely speaking, a large deviation estimate is a quantitative version of the ergodic theorem.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…Amongst the aforementioned references, [GS07] is particularly relevant to this paper, since it introduced large deviation estimates for the Dirichlet determinants on finite volume which appear as the entries of the propagator matrices M N . While [BG00] and [GS01] only consider large deviation estimates for M N , [GS07] and the present work need to go beyond the matrices and consider their entries and traces. We now describe the strategy behind the proof of Theorem 1.2 in more detail.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…The bound (2.4) for the monodromies (in an even sharper form) is in [GS01]. In [GS08] it is shown how to pass from (2.4) to (2.5); see Sections 2 and 3 of that paper.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…The avalanche principle expansion for the Dirichlet determinants. Another basic tool in this paper is the following avalanche principle; see [GS01] and [GS08].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

See 4 more Smart Citations

On resonances and the formation of gaps in the spectrum of quasi-periodic Schrödinger equations

Goldstein

Schlag

2011

Ann. Math.

Self Cite

View full text Add to dashboard Cite

We consider one-dimensional difference Schrödinger equations H(x, ω)ϕ (n) ≡ −ϕ(n − 1) − ϕ(n + 1) + V (x + nω)ϕ(n) = Eϕ(n) ,is greater than 0 for all E ∈ (E , E ) and some Diophantine ω0, then the integrated density of states is absolutely continuous for almost every ω close to ω0, as shown by the authors in earlier work. In this paper we establish the formation of a dense set of gaps in spec(H(x, ω)) ∩ (E , E ). Our approach is based on an induction on scales argument, and is therefore both constructive as well as quantitative. Resonances between eigenfunctions of one scale lead to "pre-gaps" at a larger scale. To pass to actual gaps in the spectrum, we show that these pre-gaps cannot be filled more than a finite (and uniformly bounded) number of times. To accomplish this, one relates a pre-gap to pairs of complex zeros of the Dirichlet determinants off the unit circle. Amongst other things, we establish a nonperturbative version of the co-variant parametrization of the eigenvalues and eigenfunctions via the phases in the spirit of Sinai's (perturbative) description of the spectrum via his function Λ. This allows us to relate the gaps in the spectrum with the graphs of the eigenvalues parametrized by the phase. Our infinite volume theorems hold for all Diophantine frequencies ω up to a set of Hausdorff dimension zero.

show abstract

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 62%

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…The bound (2.4) for the monodromies (in an even sharper form) is in [GS01]. In [GS08] it is shown how to pass from (2.4) to (2.5); see Sections 2 and 3 of that paper.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

“…The avalanche principle expansion for the Dirichlet determinants. Another basic tool in this paper is the following avalanche principle; see [GS01] and [GS08].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

See 3 more Smart Citations