The model magnetic Laplacian on wedges

Abstract

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