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arXiv · 2610.00733

Smooth perturbation theory and generic Cantor spectrum for multi-dimensional monotone quasiperiodic operators

Ilya Kachkovskiy·Leonid Parnovski·Roman Shterenberg
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Abstract

We consider multi-dimensional quasiperiodic Schrödinger operators with monotone sawtooth-type potentials in the perturbative localization regime. We prove that a given gap is closed iff a corresponding (via a natural gap labeling) eigenfunction vanishes at the origin. Using this, we show that for a generic choice of the sampling function $f$, the spectrum of such an operator is a Cantor set, with all possible gaps being open. We also provide a class of functions $f$ for which some gaps are closed. Finally, we observe that certain gaps are more difficult to close than the others.

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Ilya Kachkovskiy, Leonid Parnovski, Roman Shterenberg. 2026-09-30. Smooth perturbation theory and generic Cantor spectrum for multi-dimensional monotone quasiperiodic operators. https://arxiv.org/abs/2610.00733

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