The model Magnetic Laplacian on wedges

Nicolas Popoff (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

The object of this paper is model Schrödinger operators with constant magnetic fields on infinite wedges with natural boundary conditions. Such model operators play an important role in the semiclassical behavior of magnetic Laplacians on 3d domains with edges. We show that the ground energy along the wedge is lower than the energy coming from the regular part of the wedge. A consequence of this is the lower semi-continuity of the local ground energy near an edge for semi-classical Laplacians. We also show that the ground energy is Hölder with respect to the magnetic field and the wedge aperture, and even Lipschitz when the ground energy is strictly less than the energy coming from the faces. We finally provide an upper bound for the ground energy on wedges of small aperture. A few numerical computations illustrate the theoretical approach.

Mots clés: Schrödinger operator; magnetic field; singular domain; linear eigenvalue problem

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

Nicolas Popoff. The model Magnetic Laplacian on wedges. 2013. ⟨hal-00845305v4⟩