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

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.

Spectral Theory (math.SP); Mathematical Physics (math-ph); Quantum Physics (quant-ph)

arXiv:2610.00733 [math.SP]

(or arXiv:2610.00733v1 [math.SP] for this version)

https://doi.org/10.48550/arXiv.2610.00733 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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