Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances

Jean-François Bony[1]; Setsuro Fujiié[2]; Thierry Ramond[3]; Maher Zerzeri[4]

  • [1] Université Bordeaux 1 IMB (UMR CNRS 5251) 33405 Talence (France)
  • [2] University of Hyogo Graduate School of Material Science (Japan)
  • [3] Université Paris Sud 11 LMO (UMR CNRS 8628) 91405 Orsay (France)
  • [4] Université Paris 13 LAGA (UMR CNRS 7539) 93430 Villetaneuse (France)

Annales de l’institut Fourier (2011)

  • Volume: 61, Issue: 4, page 1351-1406
  • ISSN: 0373-0956

Abstract

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We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrödinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of h , and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we give an expansion for large times of the Schrödinger group in terms of these resonances.

How to cite

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Bony, Jean-François, et al. "Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances." Annales de l’institut Fourier 61.4 (2011): 1351-1406. <http://eudml.org/doc/219775>.

@article{Bony2011,
abstract = {We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrödinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of $h$, and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we give an expansion for large times of the Schrödinger group in terms of these resonances.},
affiliation = {Université Bordeaux 1 IMB (UMR CNRS 5251) 33405 Talence (France); University of Hyogo Graduate School of Material Science (Japan); Université Paris Sud 11 LMO (UMR CNRS 8628) 91405 Orsay (France); Université Paris 13 LAGA (UMR CNRS 7539) 93430 Villetaneuse (France)},
author = {Bony, Jean-François, Fujiié, Setsuro, Ramond, Thierry, Zerzeri, Maher},
journal = {Annales de l’institut Fourier},
keywords = {Schrödinger operator; quantum resonances; semiclassical analysis; resolvent estimate},
language = {eng},
number = {4},
pages = {1351-1406},
publisher = {Association des Annales de l’institut Fourier},
title = {Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances},
url = {http://eudml.org/doc/219775},
volume = {61},
year = {2011},
}

TY - JOUR
AU - Bony, Jean-François
AU - Fujiié, Setsuro
AU - Ramond, Thierry
AU - Zerzeri, Maher
TI - Spectral projection, residue of the scattering amplitude and Schrödinger group expansion for barrier-top resonances
JO - Annales de l’institut Fourier
PY - 2011
PB - Association des Annales de l’institut Fourier
VL - 61
IS - 4
SP - 1351
EP - 1406
AB - We study the spectral projection associated to a barrier-top resonance for the semiclassical Schrödinger operator. First, we prove a resolvent estimate for complex energies close to such a resonance. Using that estimate and an explicit representation of the resonant states, we show that the spectral projection has a semiclassical expansion in integer powers of $h$, and compute its leading term. We use this result to compute the residue of the scattering amplitude at such a resonance. Eventually, we give an expansion for large times of the Schrödinger group in terms of these resonances.
LA - eng
KW - Schrödinger operator; quantum resonances; semiclassical analysis; resolvent estimate
UR - http://eudml.org/doc/219775
ER -

References

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