Almost Periodicity Properties Arising From the Sodin-Yuditskii Theory

Abstract

The Kotani-Last problem and the Deift conjecture are two well known statements, which both concern the almost periodicity properties of the functions under consideration, and which have both been disproved in their original formulation, which involves Bohr/Bochner almost periodicity. To contrast these disproofs, and to further elucidate the essence of these two challenging problems, we establish positive results regarding each of them by showing that Weyl almost periodicity can be deduced whenever the Sodin-Yuditskii setup can be implemented. The latter requires that the Schrödinger operator associated with the function in question has purely absolutely continuous spectral measure and a spectrum with finite total gap length and the Parreau-Widom property. We also show that the known examples for which Bohr/Bochner almost periodicity has been shown or suggested to fail in the context of the Deift conjecture are in fact Weyl almost periodic. As a result, an accordingly modified version of the Deift conjecture remains a tantalizing open problem.

Disproving the Deift conjecture: the loss of almost periodicity
Andreia Chapouto, Rowan Killip, Monica Vişan
Séminaire Laurent Schwartz — EDP et applications4 Nov 2026Peer-reviewedCC BYFull textMathematical physicsHAL
Selfless Reduced Crossed Product C^*-Algebras Arising from Almost Periodic Actions
Syuichi Ohshima
arXiv30 Aug 2026PreprintCC BY-NC-SA 4.0Full textAnalysis & PDEsarXiv
Noncommutative weak almost periodicity and left uniform continuity over locally compact quantum groups
Ramin Faal, Amin Allah Khosravi, Hamid Reza Ebrahimi Vishki
arXiv4 Nov 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv
ACM, almost Ulrich, and Ulrich vector bundles arising from Frobenius
Jagadish Pine
arXiv20 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Representation Theory of Canonical Commutation Relations Arising from the Quantization of the Klein-Gordon Equation with an External Potential
Yasumichi Matsuzawa, Itaru Sasaki, Akito Suzuki
arXiv4 Sep 2026PreprintarXiv licenceFull textMathematical physicsarXiv

Cite

BibTeX RIS

BibTeX

@misc{damanik2026almost,
  title = {{Almost Periodicity Properties Arising From the Sodin-Yuditskii Theory}},
  author = {David Damanik and Long Li and Nicolae Strungaru},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.36403},
  eprint = {2610.36403},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.36403},
}

RIS

TY  - UNPB
TI  - Almost Periodicity Properties Arising From the Sodin-Yuditskii Theory
AU  - David Damanik
AU  - Long Li
AU  - Nicolae Strungaru
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.36403
UR  - https://arxiv.org/abs/2610.36403
AB  - The Kotani-Last problem and the Deift conjecture are two well known statements, which both concern the almost periodicity properties of the functions under consideration, and which have both been disproved in their original formulation, which involves Bohr/Bochner almost periodicity. To contrast these disproofs, and to further elucidate the essence of these two challenging problems, we establish positive results regarding each of them by showing that Weyl almost periodicity can be deduced whenever the Sodin-Yuditskii setup can be implemented. The latter requires that the Schrödinger operator associated with the function in question has purely absolutely continuous spectral measure and a spectrum with finite total gap length and the Parreau-Widom property. We also show that the known examples for which Bohr/Bochner almost periodicity has been shown or suggested to fail in the context of the Deift conjecture are in fact Weyl almost periodic. As a result, an accordingly modified version of the Deift conjecture remains a tantalizing open problem.
ER  -

Abstract

Related

Cite