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


Authors: Shu Nakamura and Alexander Pushnitski
Journal: Trans. Amer. Math. Soc. 366 (2014), 1725-1747
MSC (2010): Primary 81U20, 47F05
DOI: https://doi.org/10.1090/S0002-9947-2013-06077-1
Published electronically: December 6, 2013
MathSciNet review: 3152710
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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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Shu Nakamura
Affiliation: Graduate School of Mathematical Science, University of Tokyo, Tokyo, Japan
Email: shu@ms.u-tokyo.ac.jp

Alexander Pushnitski
Affiliation: Department of Mathematics, King’s College London, Strand, London, WC2R 2LS, United Kingdom
Email: alexander.pushnitski@kcl.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-2013-06077-1
Keywords: Shape resonances, scattering matrix, Breit-Wigner effect, semiclassical limit
Received by editor(s): March 5, 2012
Published electronically: December 6, 2013
Article copyright: © Copyright 2013 American Mathematical Society