On critical dipoles in dimensions n ≥ 3

We reconsider generalizations of Hardy's inequality corresponding to the case of (point) dipole potentials V-gamma(x) = gamma(u, x)vertical bar x vertical bar(-3), x is an element of R-n \{0}, gamma is an element of [0, infinity), u is an element of R-n, vertical bar u vertical bar = 1, n is an element of N, n >= 3. More precisely, for n >= 3, we provide an alternative proof of the existence of a critical dipole coupling constant gamma(c,n) > 0, such that for all gamma is an element of [0, gamma(c),(n)], and all u is an element of R-n, vertical bar u vertical bar =1, integral(Rn) d(n)x vertical bar(del f) (x)vertical bar(2) >= +/-gamma integral(Rn) d(n)x (u, x) vertical bar x vertical bar(-3)vertical bar f(x)vertical bar(2), f is an element of D-1(R-n) with D-1(R-n) denoting the completion of C-0(infinity)(R-n) with respect to the norm induced by the gradient. Here gamma(c,n) is sharp, that is, the largest possible such constant. Moreover, we discuss upper and lower bounds for gamma(c,n) > 0 and develop a numerical scheme for approximating gamma(c,n). This quadratic form inequality will be a consequence of the fact <([-Delta + gamma(u, x)vertical bar x vertical bar(-3)]vertical bar(C0 infinity(Rn\{0})))over bar>>= 0 if and if 0 <= gamma <= gamma(c,n) in L-2 (R-n) (with (T) over bar 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. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

The article reconsiders generalizations of Hardy's inequality regarding dipole potentials, discussing the critical coupling constant and developing a numerical approximation scheme. Additionally, it considers the case of multi-center dipole interactions.