OP-00448 Open

The bosonic Laughlin contact-interaction gap

Note: Xin Shen claims a positive answer, with c = 1/3, in “The Bosonic Laughlin Spectral Gap: A Candidate Proof”, an AI-generated note of October 2026 at https://github.com/Shencongli/bosonic-laughlin-spectral-gap. The repository is not one the registry links papers from, so the claim is neither recorded nor checked here.

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Let $\mathcal{P}_N$ be the space of symmetric complex polynomials in $z_1, \dots, z_N$ of total degree at most $N(N-1)$, with the inner product $$\langle F, G \rangle = \pi^{-N} \int_{\mathbb{C}^N} \overline{F(z)}\, G(z)\, e^{-\sum_j |z_j|^2}\, dz.$$ For $i < j$, let $P_{ij}$ be the orthogonal projection on the space of all polynomials given by $$(P_{ij} F)(z) = F\Bigl(z_1, \dots, \tfrac{z_i + z_j}{2}, \dots, \tfrac{z_i + z_j}{2}, \dots, z_N\Bigr),$$ with $(z_i + z_j)/2$ in positions $i$ and $j$. The operator $H_N = \sum_{i < j} P_{ij}$ preserves $\mathcal{P}_N$, and its kernel contains the Laughlin polynomial $\prod_{i < j} (z_i - z_j)^2$. Up to a constant factor, $H_N$ is the contact interaction projected onto the bosonic lowest Landau level, and $N(N-1)$ is the angular momentum of the Laughlin state; Lewin and Seiringer derived this model as a limit of rapidly rotating trapped Bose gases. Since $\mathcal{P}_N$ is finite-dimensional, the nonzero spectrum of $H_N$ on it is bounded away from zero for each fixed $N$.

Problem. Is there a constant $c > 0$ such that $$\inf\bigl(\sigma(H_N|_{\mathcal{P}_N}) \setminus \{0\}\bigr) \ge c$$ for every $N \ge 2$?

Source: Rougerie (2019), Appendix A, Conjecture A.1 (the bosonic case ℓ = 2), who traces it to Laughlin (1983) and Haldane (1983) and to earlier statements by Lewin and Seiringer (2009) and Rougerie, Serfaty and Yngvason (2014); stated in this form by AIM — Open Applied Problems.

Imported from github.com/MColbrook/AIM.

Classification

Primary: MP — Mathematical Physics
Secondary: SP
Keywords: AIM Open Applied Problems, Laughlin states, rotating Bose gases, spectral gaps

References

@article{rougerie2019,
  author        = {Rougerie, Nicolas},
  title         = {On the {L}aughlin function and its perturbations},
  journal       = {S{\'e}minaire Laurent Schwartz --- EDP et applications},
  year          = {2019},
  pages         = {1--17},
  note          = {S{\'e}minaire 2018--2019},
  doi           = {10.5802/slsedp.131},
  eprint        = {1906.11656},
  archivePrefix = {arXiv},
}
@article{laughlin1983,
  author  = {Laughlin, R. B.},
  title   = {Anomalous quantum {H}all effect: an incompressible quantum fluid with fractionally charged excitations},
  journal = {Phys. Rev. Lett.},
  volume  = {50},
  number  = {18},
  year    = {1983},
  pages   = {1395--1398},
  doi     = {10.1103/PhysRevLett.50.1395},
}
@article{haldane1983,
  author  = {Haldane, F. D. M.},
  title   = {Fractional quantization of the {H}all effect: a hierarchy of incompressible quantum fluid states},
  journal = {Phys. Rev. Lett.},
  volume  = {51},
  number  = {7},
  year    = {1983},
  pages   = {605--608},
  doi     = {10.1103/PhysRevLett.51.605},
}
@article{lewin2009,
  author  = {Lewin, Mathieu and Seiringer, Robert},
  title   = {Strongly correlated phases in rapidly rotating {B}ose gases},
  journal = {J. Stat. Phys.},
  volume  = {137},
  number  = {5-6},
  year    = {2009},
  pages   = {1040--1062},
  doi     = {10.1007/s10955-009-9833-y},
}
@article{rougerie2014,
  author  = {Rougerie, N. and Serfaty, S. and Yngvason, J.},
  title   = {Quantum {H}all phases and plasma analogy in rotating trapped {B}ose gases},
  journal = {J. Stat. Phys.},
  volume  = {154},
  number  = {1-2},
  year    = {2014},
  pages   = {2--50},
  doi     = {10.1007/s10955-013-0766-0},
}
@incollection{rougerie2024,
  author        = {Rougerie, Nicolas and Yngvason, Jakob},
  title         = {Quantum {H}all phases of cold {B}ose gases},
  booktitle     = {Encyclopedia of Condensed Matter Physics},
  edition       = {Second},
  publisher     = {Elsevier},
  year          = {2024},
  pages         = {640--651},
  doi           = {10.1016/B978-0-323-90800-9.00154-2},
  eprint        = {2209.14215},
  archivePrefix = {arXiv},
}
@misc{aim-laughlin-gap,
  author       = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
  title        = {{The bosonic Laughlin contact-interaction gap}},
  howpublished = {AIM --- Open Applied Problems, GitHub repository},
  year         = {2026},
  url          = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/195-laughlin-gap.md},
  note         = {Entry 195 at commit a153f46, 2026-10-13},
}

Citing this problem

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OP-00448, "The bosonic Laughlin contact-interaction gap", clopen.solutions.