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Microlocalization of resonant states and estimates of the residue of the scattering amplitude. (English) Zbl 1060.35016

Proceedings of the conference on partial differential equations, Forges-les-Eaux, France, June 2–6, 2003. Exp. Nos. I-XV. Nantes: Université de Nantes (ISBN 2-86939-207-9/pbk). Exp. No. II, 12 p. (2003).
The authors obtain some microlocal estimates of the resonant states associated to a resonance \(z_0\) of an \(h\)-differential operator. They show that the normalized resonant states are \(O(\sqrt{{|\text{Im\,} z_0|\over h}}+ h^\infty)\) outside the set of trapped trajectories and are \(O(h^\infty)\) in the incoming area of the phase space. As an application the authors show that the residue of the scattering amplitude of a Schrödinger operator is small in some directions under an estimate of the norm of the spectral projector. Finally, the authors prove such bound in some examples.
For the entire collection see [Zbl 1027.00017].

MSC:

35B34 Resonance in context of PDEs
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
35P25 Scattering theory for PDEs
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs