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

Michael Hitrik, Johannes Sjöstrand (2008)

Annales scientifiques de l'École Normale Supérieure

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...

Recent progress in the regularity theory of Fourier integrals with real and complex phases and solutions to partial differential equations

Michael Ruzhansky (2003)

Banach Center Publications

In this paper we will give a brief survey of recent regularity results for Fourier integral operators with complex phases. This will include the case of real phase functions. Applications include hyperbolic partial differential equations as well as non-hyperbolic classes of equations. An application to an oblique derivative problem is also given.

Réduction microlocale des systèmes d'opérateurs pseudo-différentiels

Jean Nourrigat (1986)

Annales de l'institut Fourier

On sait depuis 1976 qu’il existe un lien entre systèmes de champs de vecteurs réels et groupes nilpotents. On montre ici que ce phénomène s’étend aux systèmes d’opérateurs pseudo-différentiels à symboles principaux réels. Une équivalence de propriétés est conjecturée, mais seule l’une des implications est ici démontrée.

Remarks on Gårding inequalities for differential operators

Xavier Saint Raymond (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Classical Gårding inequalities such as those of Hörmander, Hörmander-Melin or Fefferman-Phong are proved by pseudo-differential methods which do not allow to keep a good control on the supports of the functions under study nor on the smoothness of the coefficients of the operator. In this paper, we show by very simple calculations that in certain special situations, the results that can be obtained directly are much better than those expected thanks to the general theory.

Resolvent and Scattering Matrix at the Maximum of the Potential

Alexandrova, Ivana, Bony, Jean-François, Ramond, Thierry (2008)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 35P25, 81U20, 35S30, 47A10, 35B38.We study the microlocal structure of the resolvent of the semiclassical Schrödinger operator with short range potential at an energy which is a unique non-degenerate global maximum of the potential. We prove that it is a semiclassical Fourier integral operator quantizing the incoming and outgoing Lagrangian submanifolds associated to the fixed hyperbolic point. We then discuss two applications of this result to describing...

Résonances près d’une énergie critique

Jean-François Bony (2001/2002)

Séminaire Équations aux dérivées partielles

Dans cet exposé, on décrit un travail effectué sous la direction de J. Sjöstrand. On prouve des majorations et des minorations du nombre de résonances d’un opérateur de Schrödinger semi-classique P = - h 2 Δ + V ( x ) dans des petits disques centrés en E 0 > 0 , une valeur critique de p ( x , ξ ) = ξ 2 + V ( x ) .

Rieffel’s pseudodifferential calculus and spectral analysis of quantum Hamiltonians

Marius Măntoiu (2012)

Annales de l’institut Fourier

We use the functorial properties of Rieffel’s pseudodifferential calculus to study families of operators associated to topological dynamical systems acted by a symplectic space. Information about the spectra and the essential spectra are extracted from the quasi-orbit structure of the dynamical system. The semi-classical behavior of the families of spectra is also studied.

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