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This paper is devoted to the symmetry and symmetry breaking properties of atwo-dimensional magnetic Schr{\"o}dinger operator involving an Aharonov-Bohmmagnetic vector potential. We investigate the symmetry properties of theoptimal potential for the corresponding magnetic Keller-Lieb-Thir-ringinequality. We prove that this potential is radially symmetric if the intensityof the magnetic field is below an explicit threshold, while symmetry is brokenabove a second threshold corresponding to a higher magnetic field. The methodrelies on the study of the magnetic kinetic energy of the wave function andamounts to study the symmetry properties of the optimal functions in a magneticHardy-Sobolev interpolation inequality. We give a quantified range of symmetryby a non-perturbative method. To establish the symmetry breaking range, weexploit the coupling of the phase and of the modulus and also obtain aquantitative result.

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