Bound states for the magnetic Neumann Laplacian in planar sectors

We study the magnetic Neumann Laplacian in an infinite planar sector of opening $\alpha\in(0,\pi)$ under a constant magnetic field. Building on earlier work by Bonnaillie-No\"el and collaborators and by Exner, Lotoreichik, and P\'erez-Obiol, we prove that the bottom of the spectrum lies strictly below the half-plane threshold for every convex sector. Consequently, $H_\alpha$ has a discrete ground-state eigenvalue for every $0<\alpha<\pi$. This resolves the bound-state problem for convex sectors, a model problem arising in the analysis of magnetic localization near corners and of the third critical field in type-II superconductivity.