Inequalities involving Aharonov-Bohm magnetic potentials in dimensions 2 and 3

Abstract

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schr{ö}dinger operators involving Aharonov-Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions two and three. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions 2 and 3.

arXiv commentReviews in Mathematical Physics, World Scientific Publishing, In press

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• Entropies, convexity, and functional inequalities
• Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces
• Properties of groundstates of nonlinear Schrödinger equations under a weak constant magnetic field
• Spectral estimates on the sphere
• Symmetry results in two-dimensional inequalities for Aharonov-Bohm magnetic fields
• Magnetic rings
• A Magnetic Contribution to the Hardy Inequality
• Interpolation inequalities and spectral estimates for magnetic operators