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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.

MSC:

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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