The 𝐿^𝑝-boundedness of wave operators for two dimensional Schrödinger operators with threshold singularities

Abstract

We generalize the recent result of Erdoğan, Goldberg and Green on the 𝐿^𝑝-boundedness of wave operators for two dimensional Schrödinger operators and prove that they are bounded in 𝐿^𝑝(ℝ²) for all 1 < 𝑝 < ∞ if and only if the Schrödinger operator possesses no 𝑝-wave threshold resonances, viz. Schrödinger equation (−Δ + 𝑉(𝑥))𝑢(𝑥) = 0 possesses no solutions which satisfy 𝑢(𝑥) = (𝑎₁ 𝑥₁ + 𝑎₂ 𝑥₂) |𝑥|⁻² + 𝑜(|𝑥|⁻¹) as |𝑥| → ∞ for an (𝑎₁, 𝑎₂) ∈ ℝ² ∖ {(0, 0)} and, otherwise, they are bounded in 𝐿^𝑝(ℝ²) for 1 < 𝑝 ≤ 2 and unbounded for 2 < 𝑝 < ∞. We present also a new proof for the known part of the result.