Let lambda(b) be the lowest eigenvalue of the magnetic Neumann Laplacian in the unit disk for a constant magnetic field of strength b>0, and let Theta_0 denote the de Gennes constant. The Helffer–Léna conjectures assert: (i) lambda is strictly increasing on (0,+∞); (ii) the normalized energies bigl(eta_n^bigr)_n∈ N at the successive crossings of the angular-momentum branches form a strictly increasing sequence, eta_n+1^>eta_n^; and (iii) the de Gennes bound holds for every field strength, lambda(b) 0.

Helffer–Léna conjectures for the magnetic Neumann Laplacian in the disk

Let $\lambda(b)$ be the lowest eigenvalue of the magnetic Neumann Laplacian in the unit disk for a constant magnetic field of strength $b>0$ , and let $\Theta_0$ denote the de Gennes constant. The Helffer–Léna conjectures assert: (i) $\lambda$ is strictly increasing on $(0,+\infty)$ ; (ii) the normalized energies $\bigl(\eta_n^*\bigr)_{n\in\mathbb N}$ at the successive crossings of the angular-momentum branches form a strictly increasing sequence, $\eta_{n+1}^*>\eta_n^*$ ; and (iii) the de Gennes bound holds for every field strength, $\lambda(b)<\Theta_0 b$ for all $b>0$ .

References

Primary source

arXiv

Additional references

Monotonicity and the de Gennes bound for the magnetic Neumann Laplacian in the disk — arXiv

ginzburglandau-theory magnetic-schrdinger-operators spectral-theory

Posted 4 weeks ago · edited 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed solved

A September 2026 preprint claims to settle all three disk conjectures, but the claim has not been independently verified.

The Helffer–Léna conjectures concern monotonicity, crossing-energy ordering, and the de Gennes bound for the magnetic Neumann Laplacian in the disk. Earlier work supplied partial results and asymptotic or numerical evidence; a new submission claims all three are proved.

Known results

A 2023 paper proved a new monotonicity property for the disk’s lowest eigenvalue, previously conjectured from numerical evidence.
A 2024 study gave asymptotic and numerical evidence and controlled the conjectures for large magnetic field, including $\lambda(\beta)=\Theta_0\beta+o(\beta)$ as $\beta\to+\infty$ .
The same study recorded the conjectures $\eta(\beta)<\Theta_0$ for $\beta>0$ and strict increase of $(\eta_n^*)_{n\in\mathbb N}$ , with the latter implying the de Gennes bound.

September 2026 claimed proof

A September 2026 arXiv submission claims proofs of the monotonicity, crossing-energy, and de Gennes-bound conjectures, which would determine the local spectral critical field for every Ginzburg–Landau parameter in the disk. The claim is unrefereed and remains unverified.

Current status (as of September 2026): all three conjectures are claimed solved by an unrefereed submission, but independent verification is absent.

Sources

No solutions have been posted yet.