The model magnetic Laplacian on wedges

We study a model Schr\"odinger operator with constan tmagnetic field on an infinite wedge with natural boundary conditions. This problem is related to the semiclassical magnetic Laplacian on 3d domains with edges. We show that the ground energy is lower than the one coming from the regular part of the wedge and is continuous with respect to the geometry. We provide an upper bound for the ground energy on wedges of small opening. Numerical computations enlighten the theoretical approach

The model magnetic Laplacian on wedges

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

On a 3d magnetic Hamiltonian with axisymmetric potential and unitary magnetic field

This study is about a magnetic Hamiltonian with axisymmetric potential in R3. The associated magnetic field is planar, unitary and non-constant. The problem reduces to a 1D family of singular Sturm-Liouville operators on the half-line. We study the associated band functions, in particular their behavior at infinity and we describe the quantum state localized in energy near the Landau levels that play the role of threshold in the spectrum. We compare our Hamiltonian to the "de Gennes" operators arising in the study of a 2D Hamiltonian with monodimensional, odd and discontinuous magnetic field. We show in particular that the ground state energy is higher in dimension 3

When the 3D magnetic Laplacian meets a curved edge in the semiclassical limit

International audienceWe study the magnetic Laplacian in the case when the Neumann boundary contains an edge. We provide complete asymptotic expansions in powers of $h^{1/4}$ of the low lying eigenpairs in the semiclassical limit $h\to 0$. In order to get our main result we establish a general method based on a normal form procedure, microlocal arguments, the Feshbach-Grushin reduction and the Born-Oppenheimer approximation