Microlocalization of resonant states and estimates of the residue of the scattering amplitude

Jean-François Bony; Laurent Michel

Journées équations aux dérivées partielles (2003)

  • page 1-12
  • ISSN: 0752-0360

Abstract

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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 ( | Im z 0 | / h + h ∞ ) outside the set of trapped trajectories and are 𝒪 ( h ∞ ) 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.

How to cite

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Bony, Jean-François, and Michel, Laurent. "Microlocalization of resonant states and estimates of the residue of the scattering amplitude." Journées équations aux dérivées partielles (2003): 1-12. <http://eudml.org/doc/93444>.

@article{Bony2003,
abstract = {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 $\{\mathcal \{O\}\} (\sqrt\{ |\mathrm \{Im\} \, z_\{0\}| /h\}$$+ h^\{\infty \})$ outside the set of trapped trajectories and are $\{\mathcal \{O\}\} (h^\{\infty \})$ 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.},
author = {Bony, Jean-François, Michel, Laurent},
journal = {Journées équations aux dérivées partielles},
keywords = {Schrödinger operator; spectral projector; -differential operator},
language = {eng},
pages = {1-12},
publisher = {Université de Nantes},
title = {Microlocalization of resonant states and estimates of the residue of the scattering amplitude},
url = {http://eudml.org/doc/93444},
year = {2003},
}

TY - JOUR
AU - Bony, Jean-François
AU - Michel, Laurent
TI - Microlocalization of resonant states and estimates of the residue of the scattering amplitude
JO - Journées équations aux dérivées partielles
PY - 2003
PB - Université de Nantes
SP - 1
EP - 12
AB - 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 ${\mathcal {O}} (\sqrt{ |\mathrm {Im} \, z_{0}| /h}$$+ h^{\infty })$ outside the set of trapped trajectories and are ${\mathcal {O}} (h^{\infty })$ 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.
LA - eng
KW - Schrödinger operator; spectral projector; -differential operator
UR - http://eudml.org/doc/93444
ER -

References

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