Microlocalization of resonant states and estimates of the residue of the scattering amplitude
Journées équations aux dérivées partielles (2003), article no. 2, 12 p.

We obtain some microlocal estimates of the resonant states associated to a resonance ${z}_{0}$ of an $h$-differential operator. More precisely, we show that the normalized resonant states are $𝒪\left(\sqrt{|\mathrm{Im}\phantom{\rule{0.166667em}{0ex}}{z}_{0}|/h}$ $+{h}^{\infty }\right)$ outside the set of trapped trajectories and are $𝒪\left({h}^{\infty }\right)$ in the incoming area of the phase space. As an application, we 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 we prove such bound in some examples.

@article{JEDP_2003____A2_0,
author = {Bony, Jean-Fran\c{c}ois and Michel, Laurent},
title = {Microlocalization of resonant states and estimates of the residue of the scattering amplitude},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {2},
publisher = {Universit\'e de Nantes},
year = {2003},
doi = {10.5802/jedp.616},
zbl = {02079437},
mrnumber = {2050588},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jedp.616/}
}
Bony, Jean-François; Michel, Laurent. Microlocalization of resonant states and estimates of the residue of the scattering amplitude. Journées équations aux dérivées partielles (2003), article  no. 2, 12 p. doi : 10.5802/jedp.616. http://www.numdam.org/articles/10.5802/jedp.616/

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