The L^p boundedness of wave operators for the Laplace operator with finite rank perturbations

2025/08/07 by Cheng, Han, Huang, Shanlin, Soffer, Avy +1
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.05533

Abstract

This paper investigates the L^p boundedness of wave operators for the Laplace operator with finite rank perturbations H=-Δ+∑_i=1^N⟨⋅ , φ_i⟩ φ_i on \R^d. For dimensions d≥ 3, we prove that the wave operators W_±(H,H₀) are bounded on L^p for the full range 1≤ p≤ ∞. This extends the work of Nier and the third author \citeNS by resolving the previously unexplored question of boundedness at the endpoint cases p=1 and p=∞. In lower dimensions d = 1, 2, we establish the L^p-boundedness of the wave operators for the first time. Furthermore, we reveal an intriguing dichotomy in the endpoint case p = 1: \beginitemize \item If ∫_ℝ^d φ_i(x) \d x = 0 holds for every 1≤ i≤ N, then the wave operators are bounded on L^p(ℝ^d) for all 1 ≤ p ≤ ∞. \item If there exists at least one i (1≤ i≤ N) such that ∫_ℝ^dφ_i(x)\d x≠0, then the wave operators remain bounded for 1 < p < ∞ and satisfy weak type (1,1) estimates, but fail to be bounded on L¹(ℝ^d). \enditemize

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