2011/01/03 by Marius Beceanu, Beceanu, Marius · 2 citations
Mathematics · #35C20 #35J10 #35P25 #47A40 #47F05 #47N50 #81U40 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1101.0502
openalex publication_date 2011/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a structure formula for the wave operators in R3 and their adjoints for a scaling-invariant class of scalar potentials V, under the assumption that zero is neither an eigenvalue, nor a resonance for -Δ+V. The formula implies the boundedness of wave operators on Lp spaces, 1 ≤ p ≤ ∞, on weighted Lp spaces, and on Sobolev spaces, as well as multilinear estimates for eitH Pc. When V decreases rapidly at infinity, we obtain an asymptotic expansion of the wave operators. The first term of the expansion is of order < y >-4, commutes with the Laplacian, and exists when V ∈ -3/2-ε L2. We also prove that the scattering operator S = W-^* W+ is an integrable combination of isometries. The proof is based on an abstract version of Wiener's theorem, applied in a new function space.