# Symmetry results in two-dimensional inequalities for Aharonov-Bohm magnetic fields

## TL;DR

This paper investigates symmetry and symmetry breaking in a two-dimensional magnetic Schrödinger operator with Aharonov-Bohm potential, identifying thresholds for magnetic field intensity that determine whether the optimal potential is symmetric or not.

## Contribution

It provides a non-perturbative analysis of symmetry properties and thresholds for symmetry breaking in magnetic inequalities involving Aharonov-Bohm fields.

## Key findings

- Optimal potential is radially symmetric below a certain magnetic field threshold.
- Symmetry breaks above a higher magnetic field threshold.
- Quantitative bounds for symmetry and symmetry breaking ranges.

## 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.