2019/12/30 by Ricardo Weder, Weder, Ricardo
Mathematics · Physics and Astronomy · #34L10 #34L25 #34L40 #47A40 #81U99 #Advanced Mathematical Physics Problems #Bounded function #Combinatorics #Eigenvalues and eigenvectors #Energy (signal processing) #FOS: Physical sciences #Identity matrix #Mathematical Physics (math-ph) #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Numerical methods in inverse problems #Physics #Quantum mechanics #Scattering #Spectral Theory in Mathematical Physics #Zero (linguistics) #math-ph #math.MP #msc:34L10 #msc:34L25 #msc:34L40 #msc:47A40 #msc:81U99
paper · pdf · doi:10.48550/arxiv.1912.12793
published in arXiv (Cornell University) (Cornell University) · The paper has been edited. Details of some proofs have been added, and the results in the boundedness of the wave operators in $L^1$ and in $L^\infty.$ are stated under slightly stronger conditions in the decay at infinity of the potential
openalex publication_date 2019/12/30 · arxiv created 2021/07/30 · arxiv updated 2021/08/03 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We prove that the wave operators for n × n matrix Schrödinger equations on the half line, with general selfadjoint boundary condition, are bounded in the spaces Lp(\mathbb R+, \mathbb Cn), 1 < p < ∞, for slowly decaying selfadjoint matrix potentials, V, that satisfy ∫0∞ (1+x) |V(x)| dx < ∞. Moreover, assuming that ∫0∞ (1+xγ) |V(x)| dx < ∞, γ> (5)/(2), and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in L1(\mathbb R+, \mathbb Cn), and in L^∞(\mathbb R+, \mathbb Cn). We also prove that the wave operators for n× n matrix Schrödinger equations on the line are bounded in the spaces Lp(\mathbb R, \mathbb Cn), 1 < p < ∞, assuming that the perturbation consists of a point interaction at the origin and of a potential, \mathcal V, that satisfies the condition ∫-∞∞ (1+|x|) |\mathcal V(x)| dx < ∞. Further, assuming that ∫-∞∞ (1+|x|γ) |\mathcal V(x)| dx < ∞, γ> (5)/(2), and that the scattering matrix is the identity at zero and infinite energy, we prove that the wave operators are bounded in L1(\mathbb R, \mathbb Cn), and in L^∞(\mathbb R, \mathbb Cn). We obtain our results for n× n matrix Schrödinger equations on the line from the results for 2n× 2n matrix Schrödinger equations on the half line.