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PreprintCC BY 4.0arXiv

(Sharp arithmetic) localization beyond transversality

  • Svetlana Jitomirskaya
  • Netanel Levi
  • Tom Stoiber
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Abstract

We prove sharp arithmetic localization for one-dimensional quasiperiodic potentials with jump discontinuities, beyond uniform transversality. Our results include every non-constant C1 monotone sampling function, even with arbitrarily large flat regions, and also certain non-monotone potentials. A key new mechanism is a symbolic unique-continuation principle: first-return words to the region where f'>0 have no common Dirichlet resonance, forcing eigenfunction mass to transfer away from flat or negatively sloped regions and recovering quantitative monotonicity of finite-volume eigenvalue branches.

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Cite

BibTeXRIS
BibTeX
@misc{jitomirskaya2026sharp,
  title = {{(Sharp arithmetic) localization beyond transversality}},
  author = {Svetlana Jitomirskaya and Netanel Levi and Tom Stoiber},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17139},
  eprint = {2610.17139},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17139},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - (Sharp arithmetic) localization beyond transversality
AU  - Svetlana Jitomirskaya
AU  - Netanel Levi
AU  - Tom Stoiber
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.17139
UR  - https://arxiv.org/abs/2610.17139
AB  - We prove sharp arithmetic localization for one-dimensional quasiperiodic potentials with jump discontinuities, beyond uniform transversality. Our results include every non-constant C^1 monotone sampling function, even with arbitrarily large flat regions, and also certain non-monotone potentials. A key new mechanism is a symbolic unique-continuation principle: first-return words to the region where f'>0 have no common Dirichlet resonance, forcing eigenfunction mass to transfer away from flat or negatively sloped regions and recovering quantitative monotonicity of finite-volume eigenvalue branches.
ER  -