2015/05/31 by Ricardo Weder · 18 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Boundary value problem #Eigenvalues and eigenvectors #Fourier transform #Hilbert space #Mathematical analysis #Mathematical physics #Mathematics #Matrix (chemical analysis) #Matrix function #Numerical methods in inverse problems #Operator (biology) #Physics #Quantum mechanics #Real line #Scattering #Scattering theory #Schrödinger equation #Spectral Theory in Mathematical Physics #Spectral theory #Symmetric matrix #math-ph #math.MP #msc:34L25 #msc:34L40 #msc:81U05 #msc:81Uxx
paper · pdf · doi:10.1063/1.4930293
published in Journal of Mathematical Physics 56(9) (American Institute of Physics) · I corrected misprints
openalex publication_date 2015/09/01 · openalex created_date 2016/06/24 · arxiv created 2018/12/20 · arxiv updated 2018/12/21 · openalex updated_date 2026/08/05
We study the stationary scattering theory for the matrix Schrödinger equation on the half line, with the most general boundary condition at the origin, and with integrable selfadjoint matrix potentials. We prove the limiting absorption principle, we construct the generalized Fourier maps, and we prove that they are partially isometric with initial space, the subspace of absolute continuity of the matrix Schrödinger operator, and final space L2((0, ∞)). We prove the existence and the completeness of the wave operators and we establish that they are given by the stationary formulae. We also construct the spectral shift function and we give its high-energy asymptotics. Furthermore, assuming that the potential also has a finite first moment, we prove a Levinson’s theorem for the spectral shift function.