Capacity & Perimeter from $\alpha$-Hermite Bounded Variation

TL;DR

This paper explores the properties of functions with $oldsymbol{ ext{α-Hermite Bounded Variation}}$, focusing on their capacity and perimeter, extending classical results from the case $oldsymbol{ ext{α=1}}$ to general $oldsymbol{ ext{α≥1}}$ in the context of the $oldsymbol{ ext{α-Hermite operator}}$.

Contribution

It introduces the concept of $ ext{α-Hermite Bounded Variation}$ spaces and analyzes their capacity and perimeter, generalizing classical bounded variation theory for the Hermite operator.

Findings

Defined $ ext{α-Hermite Bounded Variation}$ spaces.

Analyzed capacity and perimeter in these spaces.

Extended classical results to general $ ext{α}$.

Abstract

Let $\mathcal{H}_{\alpha}=\Delta-(\alpha-1)|x|^{\alpha}$ be an $[1,\infty)\ni\alpha$-Hermite operator for the hydrogen atom located at the origin in $\mathbb R^d$. In this paper, we are motivated by the classical case $\alpha=1$ to investigate the space of functions with $\alpha$-{\it Hermite Bounded Variation} and its functional capacity and geometrical perimeter.