Magnetic Laplacian in sharp three dimensional cones

Virginie Bonnaillie-Noël (1) , Monique Dauge (2) , Nicolas Popoff (3) , Nicolas Raymond (2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 Équipe EDP et Physique Mathématique

The core result of this paper is an upper bound for the ground state energy of the magnetic Laplacian with constant magnetic field on cones that are contained in a half-space. This bound involves a weighted norm of the magnetic field related to moments on a plane section of the cone. When the cone is sharp, i.e. when its section is small, this upper bound tends to 0. A lower bound on the essential spectrum is proved for families of sharp cones, implying that if the section is small enough the ground state energy is an eigenvalue. This circumstance produces corner concentration in the semi-classical limit for the magnetic Schrödinger operator when such sharp cones are involved.

Domaines: Théorie spectrale [math.SP]; Equations aux dérivées partielles [math.AP]

Virginie Bonnaillie-Noël, Monique Dauge, Nicolas Popoff, Nicolas Raymond. Magnetic Laplacian in sharp three dimensional cones. 2015. ⟨hal-01151155v1⟩