# Opening gaps in the spectrum of strictly ergodic Schrödinger operators

Artur Avila; Jairo Bochi; David Damanik

Journal of the European Mathematical Society (2012)

- Volume: 014, Issue: 1, page 61-106
- ISSN: 1435-9855

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topAvila, Artur, Bochi, Jairo, and Damanik, David. "Opening gaps in the spectrum of strictly ergodic Schrödinger operators." Journal of the European Mathematical Society 014.1 (2012): 61-106. <http://eudml.org/doc/277762>.

@article{Avila2012,

abstract = {We consider Schrödinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the possible gaps in the spectrum can be canonically labelled by an at most countable set defined purely in terms of the dynamics. Which labels actually appear depends on the choice of the sampling function; the missing labels are said to correspond to collapsed gaps. Here we show that for any collapsed gap, the sampling function may be continuously deformed so that the gap immediately opens. As a corollary, we conclude that for generic sampling functions, all gaps are open. The proof is based on the analysis of continuous SL($2,\mathbb \{R\})$ cocycles, for which we obtain dynamical results of independent interest.},

author = {Avila, Artur, Bochi, Jairo, Damanik, David},

journal = {Journal of the European Mathematical Society},

language = {eng},

number = {1},

pages = {61-106},

publisher = {European Mathematical Society Publishing House},

title = {Opening gaps in the spectrum of strictly ergodic Schrödinger operators},

url = {http://eudml.org/doc/277762},

volume = {014},

year = {2012},

}

TY - JOUR

AU - Avila, Artur

AU - Bochi, Jairo

AU - Damanik, David

TI - Opening gaps in the spectrum of strictly ergodic Schrödinger operators

JO - Journal of the European Mathematical Society

PY - 2012

PB - European Mathematical Society Publishing House

VL - 014

IS - 1

SP - 61

EP - 106

AB - We consider Schrödinger operators with dynamically defined potentials arising from continuous sampling along orbits of strictly ergodic transformations. The Gap Labeling Theorem states that the possible gaps in the spectrum can be canonically labelled by an at most countable set defined purely in terms of the dynamics. Which labels actually appear depends on the choice of the sampling function; the missing labels are said to correspond to collapsed gaps. Here we show that for any collapsed gap, the sampling function may be continuously deformed so that the gap immediately opens. As a corollary, we conclude that for generic sampling functions, all gaps are open. The proof is based on the analysis of continuous SL($2,\mathbb {R})$ cocycles, for which we obtain dynamical results of independent interest.

LA - eng

UR - http://eudml.org/doc/277762

ER -

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