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

Abstract

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.

Magnetic perturbations of anyonic and Aharonov–Bohm Schrödinger operators

“…A different approach based on quadratic form techniques was first proposed in [CO18], and later extended in [CF21,F22] to encompass generic, regular magnetic perturbations. Let us also mention that, in the single flux setting, there are classical results on Hardy-type inequalities [LW99,BDELL20] and dispersive estimates [GK14]. Another research line regards studying the AB Hamiltonian in compact domains, examining the behavior of simple eigenvalues under variations of the flux position [AFNN18,AN18] (see also [FNOS23] for similar results in configurations with many coalescing poles).…”

Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results

“…It is a remarkable feature that the use of a parabolic flow allows to bypass symmetrization techniques. This opens new directions of research in non-real valued problems (complex valued functions in quantum mechanics, in presence of magnetic fields, see for instance [23,24], or systems of equations).…”