Threshold and positive-energy bound states of the BFSS matrix model

2 papers

Proves that the undeformed relative \(\mathrm{SU}(N)\) BFSS model has exactly one normalizable zero-energy state for every finite N ≥ 2, resolving the threshold-bound-state conjecture. For \(\mathrm{SU}(2)\), a companion proves infinitely many normalizable positive-energy eigenstates with unbounded energies, contradicting the original BFSS paper's exclusion of additional bound states at N = 2.

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▶Introduction: the problem and why it mattersin preparation

▶The presented solutionin preparation

Papers 2

The unique threshold bound state of the SU(N) BFSS model

OpenAI · September 24, 2026

For every finite N ≥ 2, we prove that undeformed \(\mathop{\mathrm{SU}}\nolimits (N)\) BFSS matrix quantum mechanics has exactly one normalizable zero-energy state after removing the center of mass. This establishes the threshold-bound-state conjecture.

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@misc{OAI:The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026,
  author = {{OpenAI}},
  title = {{The unique threshold bound state of the $\mathrm{SU}(N)$ BFSS model}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026/paper.pdf}{OAI:The-unique-threshold-bound-state-of-the-SU-N-BFSS-model-September-24-2026}},
  year = {2026}
}

Positive eigenvalues of the relative SU(2) BFSS Hamiltonian

OpenAI · October 5, 2026

We prove that the relative \(\mathop{\mathrm{SU}}\nolimits (2)\) BFSS Hamiltonian has infinitely many positive eigenvalues tending to infinity, with square-integrable eigenvectors. The operator is defined by closing the gauge-invariant supercharge form. This refutes, at N = 2, the exclusion of normalizable positive-energy states stated in the original BFSS paper.

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Cite

@misc{OAI:Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026,
  author = {{OpenAI}},
  title = {{Positive eigenvalues of the relative $\mathrm{SU}(2)$ BFSS Hamiltonian}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026/positive-eigenvalues-relative-su2-bfss.pdf}{OAI:Positive-eigenvalues-of-the-relative-SU-2-BFSS-Hamiltonian-October-5-2026}},
  year = {2026}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.