On critical dipoles in dimensions n ⩾ 3

Abstract

We reconsider generalizations of Hardy's inequality corresponding to the case of (point) dipole potentials Vγ(x)=γ(u,x)|x|−3, x∈Rn﹨{0}, γ∈[0,∞), u∈Rn, |u|=1, n∈N, n⩾3. More precisely, for n⩾3, we provide an alternative proof of the existence of a critical dipole coupling constant γc,n>0, such thatfor all γ∈[0,γc,n], and all u∈Rn, |u|=1,∫Rndnx|(∇f)(x)|2⩾±γ∫Rndnx(u,x)|x|−3|f(x)|2,f∈D1(Rn) with D1(Rn) denoting the completion of C0∞(Rn) with respect to the norm induced by the gradient. Here γc,n is sharp, that is, the largest possible such constant. Moreover, we discuss upper and lower bounds for γc,n>0 and develop a numerical scheme for approximating γc,n.This quadratic form inequality will be a consequence of the fact[−Δ+γ(u,x)|x|−3]|C0∞(Rn﹨{0})‾⩾0 if and only if 0⩽γ⩽γc,n in L2(Rn) (with T‾ the operator closure of the linear operator T).We also consider the case of multicenter dipole interactions with dipoles centered on an infinite discrete set.