Opuscula Math. 39, no. 5 (2019), 611-621
https://doi.org/10.7494/OpMath.2019.39.5.611

Opuscula Mathematica

# On the imaginary part of coupling resonance points

Nurulla Azamov
Tom Daniels

Abstract. We prove for rank one perturbations that the imaginary part of a coupling resonance point is inversely proportional by a factor of $$-2$$ to the rate of change of the scattering phase, as a function of the coupling variable, evaluated at the real part of the resonance point. This equality is analogous to the Breit-Wigner formula from quantum scattering theory. For more general relatively trace class perturbations, we also give a formula for the spectral shift function in terms of coupling resonance points, non-real and real.

Keywords: scattering matrix, scattering phase, resonance point, Breit-Wigner formula.

Mathematics Subject Classification: 47A40, 47A55, 47A70.

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• Communicated by P.A. Cojuhari.
• Accepted: 2019-06-17.
• Published online: 2019-09-05.