A mathematical formulation of the Mahaux-Weidenmüller formula for the scattering matrix

Details A mathematical formulation of the Mahaux-Weidenmüller formula for the scattering matrix A mathematical formulation of the Mahaux-Weidenmüller formula for the scattering matrix

2009-03-20

arxiv.org/abs/0903.3611v3

Physics

Mathematical Physics

24 pages, 2 figures; Correction to Lemma 4.3, additional references; Elaborations in section 3; typos corrected

Scientific paper

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-Weidenm\"uller 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.

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