On critical dipoles in dimensions $n\ge 3$

Abstract

We reconsider generalizations of Hardy's inequality corresponding to the case of (point) dipole potentials $V_{\gamma }(x)=\gamma (u,x)|x{|}^{-3}$, $x\in {\mathbb{R}}^n\backslash \{0\}$, $\gamma \in [0,\infty )$, $u\in {\mathbb{R}}^n$, $|u|=1$, $n\in \mathbb{N}$, $n\ge 3$. More precisely, for $n\ge 3$, 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 γ\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.

37 pages, 1 figure, 1 table, introduction and references updated, some typos removed

References in corpus (2)

• On the best possible remaining term in the Hardy Inequality
• Bessel-Type Operators and a refinement of Hardy's inequality