Magnetic Laplacian in sharp three dimensional cones

TL;DR

This paper establishes bounds on the ground state energy of the magnetic Laplacian in sharp three-dimensional cones, revealing eigenvalue existence and corner concentration effects in the semi-classical limit.

Contribution

It provides new upper and lower bounds for the magnetic Laplacian's ground state energy in cones, linking geometry with spectral properties.

Findings

Upper bound tends to 0 for sharp cones

Eigenvalues exist for small cone sections

Corner concentration occurs in semi-classical limit

Abstract

The core result of this paper is an upper bound for the ground state energyof the magnetic Laplacian with constant magnetic field on cones that are contained in ahalf-space. This bound involves a weighted norm of the magnetic field related to momentson a plane section of the cone. When the cone is sharp, i.e. when its section is small, thisupper bound tends to 0. A lower bound on the essential spectrum is proved for familiesof sharp cones, implying that if the section is small enough the ground state energy is aneigenvalue. This circumstance produces corner concentration in the semi-classical limit forthe magnetic Schr\"odinger operator when such sharp cones are involved.