Damanik and collaborators (2007) gave the behavior for large coupling constant of the box dimension of the spectrum of a one-dimensional discrete Schrödinger operator whose potential is a Sturm sequence associated with the golden ratio. They also show that in this case the Hausdorff and box dimensions coincide (i.e. the spectrum is dimension-regular). This Note aims at giving a simpler proof of the asymptotic property result and to generalize it to the case of any Sturm potential associated with an irrational frequency whose continued fraction expansion has bounded partial quotients. Moreover, we determine the upper box dimension of the spectrum, with large coupling constant, and show that it is not dimension-regular in general.
Le comportement asymptotique pour une grande constante de couplage de la dimension du spectre d'un opérateur de Schrödinger discret dont le potentiel est une suite sturmienne associée au nombre d'or vient d'être obtenu par Damanik et al. (2007). Dans cette Note, nous donnons une démonstration plus simple de ce résultat et l'étendons au cas d'un potentiel sturmien associé à une fréquence irrationnelle dont les quotient partiels de sa décomposition en fraction continue sont bornés. Nous montrons qu'en général les dimensions de Hausdorff et de Minkowski du spectre sont différentes.
Accepted:
Published online:
Qing-Hui Liu 1; Jacques Peyrière 2, 3; Zhi-Ying Wen 3
@article{CRMATH_2007__345_12_667_0,
author = {Qing-Hui Liu and Jacques Peyri\`ere and Zhi-Ying Wen},
title = {Dimension of the spectrum of one-dimensional discrete {Schr\"odinger} operators with {Sturmian} potentials},
journal = {Comptes Rendus. Math\'ematique},
pages = {667--672},
publisher = {Elsevier},
volume = {345},
number = {12},
year = {2007},
doi = {10.1016/j.crma.2007.10.048},
language = {en},
}
TY - JOUR AU - Qing-Hui Liu AU - Jacques Peyrière AU - Zhi-Ying Wen TI - Dimension of the spectrum of one-dimensional discrete Schrödinger operators with Sturmian potentials JO - Comptes Rendus. Mathématique PY - 2007 SP - 667 EP - 672 VL - 345 IS - 12 PB - Elsevier DO - 10.1016/j.crma.2007.10.048 LA - en ID - CRMATH_2007__345_12_667_0 ER -
%0 Journal Article %A Qing-Hui Liu %A Jacques Peyrière %A Zhi-Ying Wen %T Dimension of the spectrum of one-dimensional discrete Schrödinger operators with Sturmian potentials %J Comptes Rendus. Mathématique %D 2007 %P 667-672 %V 345 %N 12 %I Elsevier %R 10.1016/j.crma.2007.10.048 %G en %F CRMATH_2007__345_12_667_0
Qing-Hui Liu; Jacques Peyrière; Zhi-Ying Wen. Dimension of the spectrum of one-dimensional discrete Schrödinger operators with Sturmian potentials. Comptes Rendus. Mathématique, Volume 345 (2007) no. 12, pp. 667-672. doi : 10.1016/j.crma.2007.10.048. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.048/
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