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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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Cite

BibTeXRIS
BibTeX
@misc{kachkovskiy2026smooth,
  title = {{Smooth perturbation theory and generic Cantor spectrum for multi-dimensional monotone quasiperiodic operators}},
  author = {Ilya Kachkovskiy and Leonid Parnovski and Roman Shterenberg},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00733},
  eprint = {2610.00733},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00733},
}
RIS
TY  - UNPB
TI  - Smooth perturbation theory and generic Cantor spectrum for multi-dimensional monotone quasiperiodic operators
AU  - Ilya Kachkovskiy
AU  - Leonid Parnovski
AU  - Roman Shterenberg
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.00733
UR  - https://arxiv.org/abs/2610.00733
AB  - 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.
ER  -