Strict angle monotonicity of magnetic wedge ground energy
For $0 < \theta < \pi$ let $W_\theta = \{(r\cos\phi, r\sin\phi) : r > 0,\ 0 < \phi < \theta\}$ be the sector of opening $\theta$, and let $H_\theta$ be the magnetic Neumann Laplacian on $W_\theta$ with constant field of strength one: the self-adjoint operator of the closed form $$q_\theta[u] = \int_{W_\theta} |(-i\nabla - A)u|^2, \qquad A(x) = \tfrac12(-x_2, x_1),$$ on $\{u \in L^2(W_\theta) : (-i\nabla - A)u \in L^2(W_\theta)\}$, with distributional derivatives. Put $\mu(\theta) = \inf \sigma(H_\theta)$. The bottom of the essential spectrum of $H_\theta$ is the de Gennes constant $\Theta_0$, the bottom of the spectrum for the half-plane. Bonnaillie-Noël proved that $\mu(\theta) < \Theta_0$, so that $\mu(\theta)$ is an eigenvalue, for $\theta \le \pi/2$, that $\mu(\theta) \sim \theta/\sqrt{3}$ as $\theta \to 0$ and $\mu(\theta) \le \theta/\sqrt{3}$, and that $\theta \mapsto \theta\mu(\theta)$ is increasing and $\theta \mapsto \mu(\theta)/\theta$ decreasing. Kachmar and Sundqvist proved $\mu(\theta) < \Theta_0$ for every $0 < \theta < \pi$ (2026).
Problem. Is $$\mu(\theta_1) < \mu(\theta_2) \qquad \text{whenever } 0 < \theta_1 < \theta_2 < \pi?$$ The field strength stays one while the opening angle varies.
Source: Bonnaillie-Noël (2005), Remark 2.6, who conjectures that the ground energy is strictly increasing in the opening angle on (0, π]. Stated by AIM — Open Applied Problems for angles in (0, π).
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- SP — Spectral Theory
- Secondary
- AP, MP
- Keywords
- AIM Open Applied Problems, corner domains, magnetic Laplacians
References
@inproceedings{bonnaillie2005jedp,
author = {Bonnaillie-No{\"e}l, Virginie},
title = {Schr{\"o}dinger operator with magnetic field in domain with corners},
booktitle = {Journ{\'e}es {\'E}quations aux d{\'e}riv{\'e}es partielles},
year = {2005},
pages = {Exp. No. II, 1--12},
doi = {10.5802/jedp.15},
}
@article{bonnaillie2005,
author = {Bonnaillie, Virginie},
title = {On the fundamental state energy for a {S}chr{\"o}dinger operator with magnetic field in domains with corners},
journal = {Asymptot. Anal.},
volume = {41},
number = {3--4},
year = {2005},
pages = {215--258},
doi = {10.3233/ASY-2005-665},
}
@misc{kachmar2026,
author = {Kachmar, Ayman and Sundqvist, Mikael},
title = {Bound states for the magnetic {N}eumann {L}aplacian in planar sectors},
year = {2026},
eprint = {2607.12600},
archivePrefix = {arXiv},
}
@misc{aim-magnetic-wedge-angle-monotonicity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Strict angle monotonicity of magnetic wedge ground energy}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/104-magnetic-wedge-angle-monotonicity.md},
note = {Entry 104 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00361, "Strict angle monotonicity of magnetic wedge ground energy", clopen.solutions.