Breit–Wigner Approximation and the Distribution¶of Resonances

For operators with a discrete spectrum, {λj2}, the counting function of λj's, N (λ), trivially satisfies N ( λ+δ ) −N ( λ−δ ) =∑jδλj((λ−δ,λ+δ]). In scattering situations the natural analogue of the discrete spectrum is given by resonances, λj∈ℂ+, and of N (λ), by the scattering phase, s(λ). The relation between the two is now non-trivial and we prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document} where ωℂ+ is the harmonic measure of the upper of half plane and δ can be taken dependent on λ. This provides a precise high energy version of the Breit–Wigner approximation, and relates the properties of s (λ) to the distribution of resonances close to the real axis.