Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2
[Tores invariants rationnels, effet tunnel dans l’espace des phases et spectres d’opérateurs non auto-adjoints en dimension 2]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 4, pp. 513-573.

Nous étudions des asymptotiques spectrales et des estimations de la résolvante des perturbations non-autoadjointes d’opérateurs h-pseudodifférentiels autoadjoints en dimension 2, en supposant que le flot classique de la partie non-perturbée soit complètement intégrable. Les contributions spectrales parvenant des tores invariants lagrangiens rationnels sont analysées. En estimant l’effet tunnel entre des tores diophantiens et rationnels, nous obtenons une description précise du spectre dans une région convenable du plan complexe spectral, sous l’hypothèse que la force de la perturbation non-autoadjointe ≫h (ou parfois ≫h 2 ) ne soit pas trop grande.

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.

DOI : 10.24033/asens.2075
Classification : 35P15, 35P20, 37J35, 37J40, 53D22, 58J37, 58J40, 70H08
Mots-clés : non-selfadjoint, eigenvalue, spectral asymptotics, resolvent, lagrangian, rational torus, diophantine torus, completely integrable, relative determinant, secular perturbation theory, phase space, tunnel effect 
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     title = {Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension $2$},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
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     publisher = {Soci\'et\'e math\'ematique de France},
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Hitrik, Michael; Sjöstrand, Johannes. Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension $2$. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 4, pp. 513-573. doi : 10.24033/asens.2075. http://archive.numdam.org/articles/10.24033/asens.2075/

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