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

Nakamura, Shu; Pushnitski, Alexander

doi:10.1090/S0002-9947-2013-06077-1

The spectrum of the scattering matrix near resonant energies in the semiclassical limit
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by Shu Nakamura and Alexander Pushnitski PDF

Trans. Amer. Math. Soc. 366 (2014), 1725-1747 Request permission

Abstract:

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 $E_\text {res}-\varepsilon$ to $E_\text {res}+\varepsilon$, where $E_\text {res}$ is a real part of a resonance and $\varepsilon$ 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.

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