Bound states for the magnetic Neumann Laplacian in planar sectors

Abstract

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-Noël and collaborators and by Exner, Lotoreichik, and Pérez-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.