Normalized crossing energies for the three-dimensional ball

Determine whether the strict monotonicity of the normalized crossing energies of consecutive angular-momentum branches for the magnetic Neumann Laplacian in the disk has an analogue for the ball in \(\mathbb{R}^3\).

Background

The paper proves that the normalized energies at successive crossings of angular-momentum branches form a strictly increasing sequence for the magnetic Neumann Laplacian in the two-dimensional disk. The authors ask whether this spectral monotonicity phenomenon persists in the rotationally symmetric three-dimensional setting of a ball in R3\mathbb{R}^3.

References

Three natural questions remain open. Does global strong diamagnetism hold for every smooth, bounded, simply connected planar domain, or at least for every convex one? Does the bound \lambda(b)<\Theta_0b hold for such domains at every field strength? For large b it follows from the two-term asymptotics of, since the maximal curvature is positive; the variational scheme of Section~\ref{sec:trial-state-proof} adapts to the large-field regime, but the treatment of a bounded range of fields uses the explicit radial trial functions. Finally, one may ask whether the monotonicity of the normalized crossing energies of Theorem~\ref{thm:crossing-quotients} has an analogue for the ball in R3.

— Monotonicity and the de Gennes bound for the magnetic Neumann Laplacian in the disk  (2609.08774 - Léna et al., 8 Sep 2026) in Section 1, subsection “Further questions”