TY - JOUR
T1 - Cantor spectrum for multidimensional quasi-periodic Schrödinger operators
AU - Helffer, Bernard
AU - Liu, Qinghui
AU - Qu, Yanhui
AU - Zhou, Qi
N1 - Publisher Copyright:
© The Author(s), 2026. Published by Cambridge University Press.
PY - 2026/6/15
Y1 - 2026/6/15
N2 - In this paper, we prove that for a dense set of irrational frequencies with positive Hausdorff dimension, the Hausdorff (and upper box) dimension of the spectrum of the critical almost Mathieu operator is positive, yet can be made arbitrarily small. As a consequence, we investigate the spectrum of a class of multidimensional quasi-periodic Schrödinger operators that exhibit a Cantor spectrum, which answers a question posed by Damanik, Fillman, and Gorodetski [24].
AB - In this paper, we prove that for a dense set of irrational frequencies with positive Hausdorff dimension, the Hausdorff (and upper box) dimension of the spectrum of the critical almost Mathieu operator is positive, yet can be made arbitrarily small. As a consequence, we investigate the spectrum of a class of multidimensional quasi-periodic Schrödinger operators that exhibit a Cantor spectrum, which answers a question posed by Damanik, Fillman, and Gorodetski [24].
UR - https://www.scopus.com/pages/publications/105042280788
U2 - 10.1017/fms.2026.10242
DO - 10.1017/fms.2026.10242
M3 - Article
AN - SCOPUS:105042280788
SN - 2050-5094
VL - 14
JO - Forum of Mathematics, Sigma
JF - Forum of Mathematics, Sigma
M1 - e90
ER -