On critical dipoles in dimensions $n\geq 3$

TL;DR

This paper establishes the existence and sharpness of a critical dipole coupling constant in higher dimensions, extending Hardy's inequality, and discusses numerical methods and bounds for this constant.

Contribution

It provides an alternative proof for the critical dipole coupling constant in dimensions n ≥ 3, including sharpness, bounds, and numerical schemes, extending Hardy's inequality to dipole potentials.

Findings

Existence of a sharp critical dipole coupling constant γ_{c,n} for n ≥ 3.

Development of a numerical scheme to compute γ_{c,n}.

Derivation of upper and lower bounds for γ_{c,n}.

Abstract

We reconsider generalizations of Hardy's inequality corresponding to the case of (point) dipole potentials $V_{\gamma}(x) = \gamma (u, x) |x|^{-3}$, $x \in \mathbb{R}^n \backslash \{0\}$, $\gamma \in [0,\infty)$, $u \in \mathbb{R}^n$, $|u|=1$, $n \in \mathbb{N}$, $n \geq 3$. More precisely, for $n \geq 3$, we provide an alternative proof of the existence of a critical dipole coupling constant $\gamma_{c,n} > 0$, such that \begin{align*} &\text{for all $\gamma \in [0,\gamma_{c,n}]$, and all $u \in \mathbb{R}^n$, $|u|=1$,} \\ &\quad \int_{\mathbb{R}^n} d^n x \, |(\nabla f)(x)|^2 \geq \pm \gamma \int_{\mathbb{R}^n} d^n x \, (u, x) |x|^{-3} |f(x)|^2, \quad f \in D^1(\mathbb{R}^n). \end{align*} with $D^1(\mathbb{R}^n)$ denoting the completion of $C_0^{\infty}(\mathbb{R}^n)$ with respect to the norm induced by the gradient. Here $\gamma_{c,n}$ is sharp, that is, the largest possible such constant, and we discuss a numerical scheme for its computation. Moreover, we discuss upper and lower bounds for $\gamma_{c,n} > 0$. We also consider the case of multicenter dipole interactions with dipoles centered on an infinite discrete set.