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Dirac Faber–Krahn conjecture

For every bounded simply connected planar domain Ω\Omega with infinite-mass boundary condition, every mass m≥0m\ge 0, and the disk B∣Ω∣B_{|\Omega|} having the same area as Ω\Omega, the first positive eigenvalue of the corresponding Dirac operator satisfies λ1+(Ω;m)≥λ1+(B∣Ω∣;m)\lambda_1^{+}(\Omega;m)\ge \lambda_1^{+}(B_{|\Omega|};m). Equality holds if and only if Ω\Omega is a disk.

References

Primary source

arXiv

Additional references

mathematical-physicspartial-differential-equationsspectral-geometry
Posted · edited

Progress summary

Refreshed
Claimed solved

A new preprint claims to prove the disk-minimization theorem in the stated planar setting, but the claim has not been independently verified.

The conjecture says that the lowest positive Dirac eigenvalue is minimized by the disk among planar domains with fixed area or perimeter. It was still described as open in 2023, but a new paper by Habib Ammari and Jiayu Qiu claims the sharp result for simply connected domains with infinite-mass boundary conditions.

Known results

  • Numerical optimization for polygons and general planar sets supplied conjectural evidence but no proof (2024).
  • A special Faber–Krahn inequality was proved for quantum-dot operators in an asymptotic boundary-condition regime (Ammari and collaborators, 2025).
  • A quantitative Payne–Weinberger-type deficit estimate was proved for bounded simply connected C2C^2 domains, but not the sharp global inequality (2026).

October 2026 claimed proof

Ammari and Qiu claim the sharp inequality and equality characterization, using spinor nonvanishing, a degree-one sphere-valued map, and planar isoperimetry. This is a specialist preprint claim and remains unverified.

Current status (as of October 2026): The stated simply connected planar infinite-mass case is claimed solved by Ammari and Qiu, while independent verification of the preprint has not been recorded.

Sources

Solutions 0

No solutions have been posted yet.