Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2

Michael Hitrik; Johannes Sjöstrand

Annales scientifiques de l'École Normale Supérieure (2008)

  • Volume: 41, Issue: 4, page 513-573
  • ISSN: 0012-9593

Abstract

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We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint h -pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational tori, we obtain an accurate description of the spectrum in a suitable complex window, provided that the strength of the non-selfadjoint perturbation ≫ h (or sometimes ≫ h 2 ) is not too large.

How to cite

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Hitrik, Michael, and Sjöstrand, Johannes. "Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension $2$." Annales scientifiques de l'École Normale Supérieure 41.4 (2008): 513-573. <http://eudml.org/doc/272212>.

@article{Hitrik2008,
abstract = {We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational tori, we obtain an accurate description of the spectrum in a suitable complex window, provided that the strength of the non-selfadjoint perturbation $\gg h$ (or sometimes $\gg h^2$) is not too large.},
author = {Hitrik, Michael, Sjöstrand, Johannes},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {non-selfadjoint; eigenvalue; spectral asymptotics; resolvent; lagrangian; rational torus; diophantine torus; completely integrable; relative determinant; secular perturbation theory; phase space; tunnel effect; pseudodifferential operators; non-selfadjoint perturbations},
language = {eng},
number = {4},
pages = {513-573},
publisher = {Société mathématique de France},
title = {Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension $2$},
url = {http://eudml.org/doc/272212},
volume = {41},
year = {2008},
}

TY - JOUR
AU - Hitrik, Michael
AU - Sjöstrand, Johannes
TI - Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension $2$
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2008
PB - Société mathématique de France
VL - 41
IS - 4
SP - 513
EP - 573
AB - We study spectral asymptotics and resolvent bounds for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part is completely integrable. Spectral contributions coming from rational invariant Lagrangian tori are analyzed. Estimating the tunnel effect between strongly irrational (Diophantine) and rational tori, we obtain an accurate description of the spectrum in a suitable complex window, provided that the strength of the non-selfadjoint perturbation $\gg h$ (or sometimes $\gg h^2$) is not too large.
LA - eng
KW - non-selfadjoint; eigenvalue; spectral asymptotics; resolvent; lagrangian; rational torus; diophantine torus; completely integrable; relative determinant; secular perturbation theory; phase space; tunnel effect; pseudodifferential operators; non-selfadjoint perturbations
UR - http://eudml.org/doc/272212
ER -

References

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