BV spaces and the perimeters related to Schrodinger operators with inverse-square potentials and applications to the rank-one theorem

Abstract

For $a\ge -(\frac{d}{2}-1{)}^2$ and $2\ddot{I}ƒ=d-2-({(d-2)}^2+4a{)}^{1/2}$, let $$\{\begin{array}{l}{\mathcal{H}}_a=-\hat{I}”+\frac{a}{{|x|}^2},\\ {\tilde{\mathcal{H}}}_{\ddot{I}}ƒ=2(-\hat{I}”+\frac{\ddot{I}{ƒ}^2}{{|x|}^2})\end{array}$$ be two Schrödinger operators with inverse-square potentials. In this paper, on the domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^d\backslash \{0\},d\ge 2,$ %apart from the origin, the ${\mathcal{H}}_a$-BV space $\mathcal{B}{\mathcal{V}}_{{\mathcal{H}}_a}(\hat{I}\text{c◯})$ and the ${\tilde{\mathcal{H}}}_{\ddot{I}}ƒ$-BV space $\mathcal{B}{\mathcal{V}}_{{\tilde{\mathcal{H}}}_{\ddot{I}}ƒ}(\hat{I}\text{c◯})$ related to ${\mathcal{H}}_a$ and ${\tilde{\mathcal{H}}}_{\ddot{I}}ƒ$ are introduced, respectively. We investigate a series of basic properties of $\mathcal{B}{\mathcal{V}}_{{\mathcal{H}}_a}(\hat{I}\text{c◯})$ and $\mathcal{B}{\mathcal{V}}_{{\tilde{\mathcal{H}}}_{\ddot{I}}ƒ}(\hat{I}\text{c◯})$. Furthermore, we prove that ${\tilde{\mathcal{H}}}_{\ddot{I}}ƒ$-restricted BV functions can be characterized equivalently via their subgraphs. As applications, we derive the rank-one theorem for ${\tilde{\mathcal{H}}}_{\ddot{I}}ƒ$-restricted BV functions.