BV spaces and the perimeters related to Schrödinger operators with inverse-square potentials and applications to the rank-one theorem. (September 2022) - Han, Yang; Huang, Jizheng; Li, Pengtao; Liu, Yu
Abstract: For a ≥ − ( d 2 − 1 ) 2 and 2 σ = d − 2 − ( ( d − 2 ) 2 + 4 a ) 1 2, let H a = − Δ + a | x | 2, H ˜ σ = 2 ( − Δ + σ 2 | x | 2 ) be two Schrödinger operators with inverse-square potentials. In this paper, on the domain Ω ⊂ R d ∖ 0 , d ≥ 2, the H a -BV space B V H a ( Ω ) and the H ˜ σ -BV space B V H ˜ σ ( Ω ) related to H a and H ˜ σ are introduced, respectively. We investigate a series of basic properties of B V H a ( Ω ) and B V H ˜ σ ( Ω ) . Furthermore, we prove that H ˜ σ -restricted BV functions can be characterized equivalently via their subgraphs. As applications, we derive the rank-one theorem for H ˜ σ -restricted BV functions.