On the Semiclassical Magnetic Laplacian and Connected Topics

From the Laplacian with variable magnetic field to the electric Laplacian in the semiclassical limit

We consider a twisted magnetic Laplacian with Neumann condition on a smooth and bounded domain of R2 in the semiclassical limit h → 0. Under generic assumptions, we prove that the eigenvalues admit a complete asymptotic expansion in powers of h1/4.

Semiclassical Analysis with Vanishing Magnetic Fields

We analyze the 2D magnetic Laplacian in the semiclassical limit in the case when the magnetic field vanishes along a smooth curve. In particular, we prove local and microlocal estimates for the eigenfunctions and a complete asymptotic expansion of the eigenpairs in powers of h.

The model Magnetic Laplacian on wedges

We study a model Schrödinger operator with constant magnetic 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 e...

When the 3D Magnetic Laplacian Meets a Curved Edge in the Semiclassical Limit

We study the magnetic Laplacian in the case when the Neumann boundary contains an edge. We provide complete asymptotic expansions in powers of h of the low lying eigenpairs in the semiclassical limit h→ 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.

On the semiclassical 3D Neumann Laplacian with variable magnetic field