In dimensions 2 and 3, the sharp constant in the second-order Hydrogen uncertainty principle is strictly smaller than previously conjectured; a weighted extension has sharp radial extremals.
The sharp second order Caffareli-Kohn-Nirenberg inequality and stability estimates for the sharp second order uncertainty principle
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abstract
In this paper we prove a class of second order Caffarelli-Kohn-Nirenberg inequalities which contains the sharp second order uncertainty principle recently established by Cazacu, Flynn and Lam \cite{CFL2020} as a special case. We also show the sharpness of our inequalities for several classes of parameters. Finally, we prove two stability versions of the sharp second order uncertainty principle of Cazacu, Flynn and Lam by showing that the difference of both sides of the inequality controls the distance to the set of extremal functions in $L^2$ norm of gradient of functions.
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On the extremal functions of second order uncertainty principles: symmetry and symmetry breaking
In dimensions 2 and 3, the sharp constant in the second-order Hydrogen uncertainty principle is strictly smaller than previously conjectured; a weighted extension has sharp radial extremals.