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PreprintCC BY 4.0arXiv

Poisson statistics for one-dimensional Anderson--Bernoulli Schrödinger operators

  • Yi Han
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Abstract

We prove Poisson local eigenvalue statistics for one-dimensional Schrödinger operators with nondegenerate Bernoulli potentials at any fixed nonzero coupling. After unfolding by the integrated density of states (IDS), the local eigenvalue process converges to a Poisson point process of unit intensity at every energy in the interior of the almost-sure spectrum. The theorem also holds for bounded nonconstant iid potentials, provided the IDS is continuous and strictly increasing near the reference energy. If the IDS also has a finite positive derivative at that energy, the usual linear normalization gives the same limit. The proof uses a phase coordinate with exact IDS drift to obtain a uniform exponential crossing law and joint Poisson counts in disjoint energy windows.

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Cite

BibTeXRIS
BibTeX
@misc{han2026poisson,
  title = {{Poisson statistics for one-dimensional Anderson--Bernoulli Schrödinger operators}},
  author = {Yi Han},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21268},
  eprint = {2610.21268},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21268},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Poisson statistics for one-dimensional Anderson--Bernoulli Schrödinger operators
AU  - Yi Han
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21268
UR  - https://arxiv.org/abs/2610.21268
AB  - We prove Poisson local eigenvalue statistics for one-dimensional Schrödinger operators with nondegenerate Bernoulli potentials at any fixed nonzero coupling. After unfolding by the integrated density of states (IDS), the local eigenvalue process converges to a Poisson point process of unit intensity at every energy in the interior of the almost-sure spectrum. The theorem also holds for bounded nonconstant iid potentials, provided the IDS is continuous and strictly increasing near the reference energy. If the IDS also has a finite positive derivative at that energy, the usual linear normalization gives the same limit. The proof uses a phase coordinate with exact IDS drift to obtain a uniform exponential crossing law and joint Poisson counts in disjoint energy windows.
ER  -