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PreprintarXiv licencearXiv

Lenses make many points: Pure point spectrum disproves Simon's L2 conjecture

  • Simon Becker
  • Mengxuan Yang
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Abstract

For every d≥2 and 0≤σ<1/3, we construct with LLM-assistance a smooth bounded nonnegative lens-type potential V with arbitrarily small |V|∞+∫ℝd|x|1−d+σ|V(x)|2 dx, such that −Δ+V has only pure point spectrum in (0,∞). Thus, we show that Barry Simon's L2 conjecture, Problem 8 in "Schrödinger operators in the twenty-first century", is false.

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Cite

BibTeXRIS
BibTeX
@misc{becker2026lenses,
  title = {{Lenses make many points: Pure point spectrum disproves Simon's $L^2$ conjecture}},
  author = {Simon Becker and Mengxuan Yang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31637},
  eprint = {2610.31637},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31637},
}
RIS
TY  - UNPB
TI  - Lenses make many points: Pure point spectrum disproves Simon's L^2 conjecture
AU  - Simon Becker
AU  - Mengxuan Yang
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31637
UR  - https://arxiv.org/abs/2610.31637
AB  - For every d≥2 and 0≤σ<1/3, we construct with LLM-assistance a smooth bounded nonnegative lens-type potential V with arbitrarily small |V|_∞+∫_ℝ^d|x|^1−d+σ|V(x)|^2 dx, such that −Δ+V has only pure point spectrum in (0,∞). Thus, we show that Barry Simon's L^2 conjecture, Problem 8 in "Schrödinger operators in the twenty-first century", is false.
ER  -