Magnetic Laplacian in sharp three dimensional cones
Résumé
The core result of this paper is an upper bound for the ground state energy
of the magnetic Laplacian with constant magnetic field on cones that are contained in a
half-space. This bound involves a weighted norm of the magnetic field related to moments
on a plane section of the cone. When the cone is sharp, i.e. when its section is small, this
upper bound tends to 0. A lower bound on the essential spectrum is proved for families
of sharp cones, implying that if the section is small enough the ground state energy is an
eigenvalue. This circumstance produces corner concentration in the semi-classical limit for
the magnetic Schrödinger operator when such sharp cones are involved.
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