Cantor spectrum for multidimensional quasi-periodic Schrödinger operators

2025/06/04 by Helffer, Bernard, Liu, Qinghui, Qu, Yanhui +1 · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Metric Geometry (math.MG) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2506.03577

Abstract

In this paper, we investigate the spectrum of a class of multidimensional quasi-periodic Schrödinger operators that exhibit a Cantor spectrum, which provides a resolution to a question posed by Damanik, Fillman, and Gorodetski \citeDFG. Additionally, 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.

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Cited by

• Log-Hausdorff multifractality of the absolutely continuous spectral measure of the almost Mathieu operator

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