TY - JOUR

T1 - ON THE ABOMINABLE PROPERTIES OF THE ALMOST MATHIEU OPERATOR WITH WELL-APPROXIMATED FREQUENCIES

AU - Avila, Artur

AU - Last, Yoram

AU - Shamis, Mira

AU - Zhou, Qi

N1 - Publisher Copyright:
© 2024 Duke University Press. All rights reserved.

PY - 2024/3/15

Y1 - 2024/3/15

N2 - We show that some spectral properties of the almost Mathieu operator with frequency well approximated by rationals can be as poor as at all possible in the class of all one-dimensional discrete Schrödinger operators. For the case of critical coupling, we show that the Hausdorff measure of the spectrum may vanish (for appropriately chosen frequencies) whenever the gauge function tends to zero faster than logarithmically. For arbitrary coupling, we show that modulus of continuity of the integrated density of states can be arbitrary close to logarithmic; we also prove a similar result for the Lyapunov exponent as a function of the spectral parameter. Finally, we show that (for any coupling) there exist frequencies for which the spectrum is not homogeneous in the sense of Carleson, and, moreover, fails the Parreau-Widom condition. The frequencies for which these properties hold are explicitly described in terms of the growth of the denominators of the convergents.

AB - We show that some spectral properties of the almost Mathieu operator with frequency well approximated by rationals can be as poor as at all possible in the class of all one-dimensional discrete Schrödinger operators. For the case of critical coupling, we show that the Hausdorff measure of the spectrum may vanish (for appropriately chosen frequencies) whenever the gauge function tends to zero faster than logarithmically. For arbitrary coupling, we show that modulus of continuity of the integrated density of states can be arbitrary close to logarithmic; we also prove a similar result for the Lyapunov exponent as a function of the spectral parameter. Finally, we show that (for any coupling) there exist frequencies for which the spectrum is not homogeneous in the sense of Carleson, and, moreover, fails the Parreau-Widom condition. The frequencies for which these properties hold are explicitly described in terms of the growth of the denominators of the convergents.

UR - http://www.scopus.com/inward/record.url?scp=85190978977&partnerID=8YFLogxK

U2 - 10.1215/00127094-2023-0022

DO - 10.1215/00127094-2023-0022

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AN - SCOPUS:85190978977

SN - 0012-7094

VL - 173

SP - 603

EP - 672

JO - Duke Mathematical Journal

JF - Duke Mathematical Journal

IS - 4

ER -