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

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Date
2019Publisher city
ParisLink to item file
https://hal.archives-ouvertes.fr/hal-02003872Dewey
AnalyseSujet
Hardy-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 constantsJournal issue
Communications in Mathematical PhysicsPublication date
2019Article pages
17Publisher
SpringerCollections
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Show full item recordAuthor
Bonheure, Denis
Dolbeault, Jean
Esteban, Maria J.
Laptev, Ari
Loss, Michael
Type
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.Related items
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