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.
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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.