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Symmetry Results in Two-Dimensional Inequalities for Aharonov–Bohm Magnetic Fields

par Bonheure, Denis ;Dolbeault, Jean;Esteban, María José;Laptev, Ari;Loss, Michael
Référence Communications in Mathematical Physics, 375, 3, page (2017-2087)
Publication Publié, 2020-01-01

Article révisé par les pairs

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Résumé :

This paper is devoted to the symmetry and symmetry breaking properties of a two-dimensional magnetic Schrödinger operator involving an Aharonov–Bohm magnetic vector potential. We investigate the symmetry properties of the optimal potential for the corresponding magnetic Keller–Lieb–Thirring 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.