Symmetry results in two-dimensional inequalities for Aharonov-Bohm magnetic fields
Abstract
This paper is devoted to the symmetry and symmetry breaking properties of a two-dimensional magnetic Schr{\"o}dinger operator involving an Aharonov-Bohm magnetic vector potential. We investigate the symmetry properties of the optimal potential for the corresponding magnetic Keller-Lieb-Thir-ring inequality. We prove that this potential is radially symmetric if the intensity of the magnetic field is below an explicit threshold, while symmetry is broken above a second threshold corresponding to a higher magnetic field. The method relies on the study of the magnetic kinetic energy of the wave function and amounts to study the symmetry properties of the optimal functions in a magnetic Hardy-Sobolev interpolation inequality. We give a quantified range of symmetry by a non-perturbative method. To establish the symmetry breaking range, we exploit the coupling of the phase and of the modulus and also obtain a quantitative result.
Keywords
Cite
@article{arxiv.1902.01065,
title = {Symmetry results in two-dimensional inequalities for Aharonov-Bohm magnetic fields},
author = {Denis Bonheure and Jean Dolbeault and Maria J. Esteban and Ari Laptev and Michael Loss},
journal= {arXiv preprint arXiv:1902.01065},
year = {2019}
}