Symmetry results in two-dimensional inequalities for Aharonov-Bohm magnetic fields
Bonheure, Denis; Dolbeault, Jean; Esteban, Maria J.; Laptev, Ari; Loss, Michael (2019), Symmetry results in two-dimensional inequalities for Aharonov-Bohm magnetic fields, Communications in Mathematical Physics, p. 17. 10.1007/s00220-019-03560-y
TypeArticle accepté pour publication ou publié
External document linkhttps://hal.archives-ouvertes.fr/hal-02003872
Journal nameCommunications in Mathematical Physics
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Abstract (EN)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-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.
Subjects / KeywordsHardy-Sobolev inequalities; Caffarelli-Kohn-Nirenberg inequalities; magnetic rings; magnetic Schrödinger operator; Aharonov-Bohm magnetic potential; radial symmetry; symmetry breaking; magnetic Hardy-Sobolev inequality; magnetic interpolation inequality; optimal constants
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Bonheure, Denis; Dolbeault, Jean; Esteban, Maria J.; Laptev, Ari; Loss, Michael (2019) Document de travail / Working paper
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