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PreprintCC BY 4.0arXiv

Sharp Logarithmic Quantum Dynamics for Quasiperiodic Schrödinger Operators

  • Wencai Liu
  • Xueyin Wang
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Abstract

Dynamical localization requires all position moments of a quantum wavepacket to remain bounded in time, but for quasiperiodic Schrödinger operators such bounds are generally not uniform in phase. In the positive Lyapunov exponent regime, the best known phase-uniform estimates instead grow on a logarithmic scale. We prove that both the logarithmic scale and the dependence on the moment order are sharp for a class of one-frequency quasiperiodic Schrödinger operators with even potentials. Our main ingredient is a reflective version of semi-uniformly localized eigenfunctions, adapted to the two localization centers forced by a completely resonant phase, from which we obtain matching logarithmic lower bounds along sequences of times.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026sharp,
  title = {{Sharp Logarithmic Quantum Dynamics for Quasiperiodic Schrödinger Operators}},
  author = {Wencai Liu and Xueyin Wang},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29339},
  eprint = {2608.29339},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29339},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Sharp Logarithmic Quantum Dynamics for Quasiperiodic Schrödinger Operators
AU  - Wencai Liu
AU  - Xueyin Wang
PY  - 2026
DA  - 2026/08/29
PB  - arXiv
DO  - 10.48550/arxiv.2608.29339
UR  - https://arxiv.org/abs/2608.29339
AB  - Dynamical localization requires all position moments of a quantum wavepacket to remain bounded in time, but for quasiperiodic Schrödinger operators such bounds are generally not uniform in phase. In the positive Lyapunov exponent regime, the best known phase-uniform estimates instead grow on a logarithmic scale. We prove that both the logarithmic scale and the dependence on the moment order are sharp for a class of one-frequency quasiperiodic Schrödinger operators with even potentials. Our main ingredient is a reflective version of semi-uniformly localized eigenfunctions, adapted to the two localization centers forced by a completely resonant phase, from which we obtain matching logarithmic lower bounds along sequences of times.
ER  -