Ground Energy of the Magnetic Laplacian in Polyhedral Bodies

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

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 CPT - Centre de Physique Théorique - UMR 7332

The asymptotic behavior of the first eigenvalues of magnetic Laplacian operators with large magnetic fields and Neumann realization in polyhedral domains is characterized by a hierarchy of model problems. We investigate properties of the model problems (continuity, semi-continuity, existence of generalized eigenfunctions). We prove estimates for the remainders of our asymptotic formula. Lower bounds are obtained with the help of a classical IMS partition based on adequate coverings of the polyhedral domain, whereas upper bounds are established by a novel construction of quasimodes, qualified as sitting or sliding according to spectral properties of local model problems.

Mots clés: Magnetic Laplacian; Schrödinger operator with magnetic field; polyhedral domains; singular chains; semi-classical parameter

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]

Virginie Bonnaillie-Noël, Monique Dauge, Nicolas Popoff. Ground Energy of the Magnetic Laplacian in Polyhedral Bodies. 2013. ⟨hal-00864272v2⟩