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

For $a \ge - {( \frac{{d}}{2}- 1)^2} $ and $2\sigma= {{d - 2}}-( {{{(d - 2)}^2} + 4a})^{1/2}$, let $$\begin{cases}\mathcal{H}_{a}= - \Delta + \frac{a} {{{{ | x |}^2}}},\\ \mathcal{\widetilde{H}}_{\sigma}= 2\big( { - \Delta + \frac{{{\sigma ^2}}} {{{{ | x |}^2}}}}\big)\end{cases}$$ be two Schr\"odinger operators with inverse-square potentials. In this paper, on the domain $\Omega \subset {\mathbb {R}^d}\backslash \{ 0\}, d\geq 2,$ %apart from the origin, the ${\mathcal{H} _a}$-BV space $\mathcal{B} {\mathcal{V} _{{\mathcal{H} _a}}}(\Omega )$ and the ${\mathcal{\widetilde{H}}_{\sigma}}$-BV space $\mathcal{B} {\mathcal{V} _{{\mathcal{\widetilde H} _\sigma}}}(\Omega )$ related to $\mathcal{H}_{a}$ and $\mathcal{\widetilde{H}}_{\sigma}$ are introduced, respectively. We investigate a series of basic properties of $\mathcal{B} {\mathcal{V} _{{\mathcal{H} _a}}}(\Omega )$ and $\mathcal{B} {\mathcal{V} _{{\mathcal{\widetilde H} _{\sigma}}}}(\Omega )$. Furthermore, we prove that ${\mathcal{\widetilde{H}}_{\sigma}}$-restricted BV functions can be characterized equivalently via their subgraphs. As applications, we derive the rank-one theorem for ${\mathcal{\widetilde{H}}_{\sigma}}$-restricted BV functions.