Critical Dipoles in Dimensions n ≥ 3

We consider generalizations of Hardy's inequality corresponding to the case of (point) dipole potentials in three or more spatial dimensions. More precisely, we prove the existence of a critical dipole coupling constant γ>0 such that a Hardy-type inequality holds for infinitely differentiable functions which are compactly supported in n-dimensional Euclidean space away from the origin. This coupling constant is optimal, that is, the largest possible such constant, and we discuss a numerical scheme for its computation.

The quadratic form inequality will be a consequence of the fact that the L2-closure of the dipole operator is bounded from below by zero if and only if the coupling constant ranges from 0 to the critical γ.