Monotonicity and the de Gennes bound for the magnetic Neumann Laplacian in the disk
Abstract
We consider the lowest eigenvalue λ(b) of the magnetic Neumann Laplacian in the unit disk, for a constant magnetic field of strength b>0. We prove that λ is strictly increasing on (0,+∞). This means that strong diamagnetism holds at every field strength, and not only at large ones. We also show that the normalized energies at the successive crossings of angular-momentum branches form a strictly increasing sequence; combined with the strong-field asymptotics, this gives the global bound λ(b)<Θ0 b, where Θ0 is the de Gennes constant. These results settle the three conjectures formulated by Helffer and Léna for the disk. As a consequence, the local, or spectral, critical field HC3loc of Ginzburg--Landau theory is, in the disk, uniquely determined for every value of the Ginzburg--Landau parameter, and not only for large ones. We also give a second proof of the bound λ(b)<Θ0 b, independent of the first and of the results of Helffer and Léna, by a direct variational method: trial states built from the de Gennes ground state for large fields, constant trial states for small fields, and, on the remaining bounded field interval, finite-dimensional spaces of polynomial trial states certified by finitely many exact computations in rational arithmetic. That proof uses no asymptotic input. It yields in addition an explicit upper bound for λ(b), valid above an explicit field strength, whose two leading terms are those of the strong-field asymptotics.
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BibTeX
@misc{lena2026monotonicity,
title = {{Monotonicity and the de Gennes bound for the magnetic Neumann Laplacian in the disk}},
author = {Corentin Léna and Mikael Sundqvist},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08774},
eprint = {2609.08774},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08774},
}
RIS
TY - UNPB
TI - Monotonicity and the de Gennes bound for the magnetic Neumann Laplacian in the disk
AU - Corentin Léna
AU - Mikael Sundqvist
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08774
UR - https://arxiv.org/abs/2609.08774
AB - We consider the lowest eigenvalue λ(b) of the magnetic Neumann Laplacian in the unit disk, for a constant magnetic field of strength b>0. We prove that λ is strictly increasing on (0,+∞). This means that strong diamagnetism holds at every field strength, and not only at large ones. We also show that the normalized energies at the successive crossings of angular-momentum branches form a strictly increasing sequence; combined with the strong-field asymptotics, this gives the global bound λ(b)<Θ_0 b, where Θ_0 is the de Gennes constant. These results settle the three conjectures formulated by Helffer and Léna for the disk. As a consequence, the local, or spectral, critical field H_C_3^loc of Ginzburg--Landau theory is, in the disk, uniquely determined for every value of the Ginzburg--Landau parameter, and not only for large ones. We also give a second proof of the bound λ(b)<Θ_0 b, independent of the first and of the results of Helffer and Léna, by a direct variational method: trial states built from the de Gennes ground state for large fields, constant trial states for small fields, and, on the remaining bounded field interval, finite-dimensional spaces of polynomial trial states certified by finitely many exact computations in rational arithmetic. That proof uses no asymptotic input. It yields in addition an explicit upper bound for λ(b), valid above an explicit field strength, whose two leading terms are those of the strong-field asymptotics.
ER -