The spectrum of the scattering matrix near resonant energies in the semiclassical limit

The object of study in this paper is the on-shell scattering matrix S(E) of the Schrödinger operator with the potential satisfying assumptions typical in the theory of shape resonances. We study the spectrum of S(E) in the semiclassical limit when the energy parameter E varies from E res - ε to E res + ε, where E res is a real part of a resonance and ε is sufficiently small. The main result of our work describes the spectral flow of the scattering matrix through a given point on the unit circle. This result is closely related to the Breit-Wigner effect.