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

No reviews yet.

No comments yet.