2009/03/20 by T. J. Christiansen, Christiansen, T. J., Maciej Zworski +2
Mathematics · Physics and Astronomy · #35P25 #58J50 #81U20 #FOS: Mathematics #FOS: Physical sciences #Laser-Matter Interactions and Applications #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #math-ph #math.MP #math.SP #msc:35P25 #msc:58J50 #msc:81U20
paper · pdf · doi:10.48550/arxiv.0903.3611
24 pages, 2 figures; Correction to Lemma 4.3, additional references; Elaborations in section 3; typos corrected
openalex publication_date 2009/03/20 · arxiv created 2009/07/31 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The purpose of this note is to give a mathematical explanation of a formula for the scattering matrix for a manifold with infinite cylindrical ends or a waveguide. This formula, which is well known in the physics literature, is sometimes referred to as the Mahaux-Weidenmüller formula. We show that a version of this formula given below gives the standard scattering matrix used in the mathematics literature. We also show that the finite rank approximation of the interaction matrix gives an approximation of the scattering matrix with errors inversely proportional to a fixed dimension-dependent power of the rank. A simple example shows that this estimate is optimal.