Breit-Wigner approximation and the distribution of resonances. (English) Zbl 0936.47004

Summary: For operators with a discrete spectrum, \(\{\lambda^2_j\}\), the counting function of \(\lambda_j\)’s, \(N(\lambda)\), trivially satisfies \(N(\lambda+ \delta)- N(\lambda- \delta)= \sum_j\delta_{\lambda_j}((\lambda- \delta,\lambda+\delta])\). In scattering situations the natural analogue of the discrete spectrum is given by resonances, \(\lambda_j\in \mathbb{C}_+\), and of \(N(\lambda)\), by the scattering phase, \(s(\lambda)\). The relation between the two is now nontrivial and we prove that \[ s(\lambda+ \delta)- s(\lambda- \delta)= \sum_{|\lambda_j- \lambda|< \varepsilon} \omega_{\mathbb{C}_+}(\lambda_j, [\lambda-\delta, \lambda+\delta])+{\mathcal O}(\delta) \lambda^{n- 1}, \] where \(\omega_{\mathbb{C}_+}\) is the harmonic measure of the upper of half plane and \(\delta\) can be taken dependent on \(\lambda\). This provides a precise high energy version of the Breit-Wigner approximation, and relates the properties of \(s(\lambda)\) to the distribution of resonances close to the real axis.


47A40 Scattering theory of linear operators
81U99 Quantum scattering theory
47F05 General theory of partial differential operators
47N50 Applications of operator theory in the physical sciences
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