Lagrangian tori and spectra for non-selfadjoint operators (based on joint works with J. Sjöstrand and S. Vũ Ngọc)

Hitrik, Michael

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2005-2006), Exposé no. 24, 14 p.

[1] L. Boutet de Monvel and J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegö, Journées Équations aux Dérivées Partielles de Rennes (1975), Astérisque 34–35 (1976), 123–164. | Numdam | MR | Zbl

[2] H. Broer and G. B. Huitema, A proof of the isoenergetic KAM theorem from the “ordinary” one, Journal of Differential Equations, 90 (1991), 52–60. | Zbl

[3] N. Burq and M. Hitrik, Energy decay for damped wave equations on partially rectangular domains, Math. Research Letters, to appear. | MR | Zbl

[4] Y. Colin de Verdière, Quasi-modes sur les variétés riemanniennes, Inv. Math. 43 (1977), 15–52. | MR | Zbl

[5] Y. Colin de Verdière, Sur le spectre des opérateurs elliptiques a bicaractéristiques toutes periodiques, Comment Math. Helv. 54 (1979), 508-522. | MR | Zbl

[6] Y. Colin de Verdière, Méthodes semi-classiques et théorie spectrale, Cours de DEA, Institut Fourier, 1992.

[7] Y. Colin de Verdière and S. Vũ Ngọc, Singular Bohr-Sommerfeld rules for 2D integrable systems, Ann. Sci. École Norm. Sup. 36 (2003), 1–55. | Numdam | MR | Zbl

[8] N. Dencker, J. Sjöstrand, and M. Zworski, Pseudospectra of semiclassical (pseudo)differential operators, Comm. Pure Appl. Math. 57 (2004), 384–415. | MR | Zbl

[9] M. Dimassi and J. Sjöstrand, Spectral asymptotics in the semi-classical limit, Cambridge University Press, Cambridge, 1999. | MR | Zbl

[10] I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18 American Mathematical Society, Providence, R.I. 1969. | MR | Zbl

[11] M. Hitrik, Eigenfrequencies for damped wave equations on Zoll manifolds, Asymptot. Analysis, 31 (2002), 265–277. | MR | Zbl

[12] M. Hitrik and J. Sjöstrand, Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions I, Ann. Henri Poincaré 5 (2004), 1–73. | MR | Zbl

[13] M. Hitrik and J. Sjöstrand, Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions II. Vanishing averages, Comm. Partial Differential Equations 30 (2005), 1065–1106. | MR | Zbl

[14] M. Hitrik and J. Sjöstrand, Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions III a. One branching point, submitted. | Zbl

[15] M. Hitrik and J. Sjöstrand, Rational tori and spectra for non-selfadjoint operators in dimension 2, in preparation.

[16] M. Hitrik, J. Sjöstrand, and S. Vũ Ngọc, Diophantine tori and spectral asymptotics for non-selfadjoint operators, American Journal of Mathematics, to appear.

[17] V. Ivrii, Microlocal analysis and precise spectral asymptotics, Springer–Verlag, Berlin, 1998. | MR | Zbl

[18] N. Kaidi and P. Kerdelhué, Forme normale de Birkhoff et résonances, Asymptot. Analysis, 23 (2000), 1–21. | MR | Zbl

[19] H. Koch and D. Tataru, On the spectrum of hyperbolic semigroups, Comm. Partial Differential Equations 20 (1995), 901–937. | MR | Zbl

[20] V. F. Lazutkin, KAM theory and semiclassical approximations to eigenfunctions. With an Addendum by A. I. Shnirelman. Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 1993. | MR | Zbl

[21] G. Lebeau, Equation des ondes amorties. Algebraic and geometric methods in mathematical physics (Kaciveli 1993), 73–109, Math. Phys. Stud., 19, Kluwer Acad. Publ., Dordrecht, 1996. | MR | Zbl

[22] A. J. Lichtenberg and M. A. Lieberman, Regular and chaotic dynamics, Second edition. Springer–Verlag, New York, 1992. | MR | Zbl

[23] A. S. Markus, Introduction to the spectral theory of polynomial operator pencils, Translations of Mathematical Monographs 71, American Mathematical Society, Providence RI, 1998. | MR | Zbl

[24] A. Melin and J. Sjöstrand, Determinats of pseudodifferential operators and complex deformations of phase space, Methods and Applications of Analysis 9 (2002), 177–238. | MR | Zbl

[25] A. Melin and J. Sjöstrand, Bohr-Sommerfeld quantization condition for non-selfadjoint operators in dimension 2, Astérisque 284 (2003), 181–244. | Numdam | MR | Zbl

[26] G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms. I. Birkhoff normal forms, Ann. Henri Poincaré 1 (2000), 223–248. | MR | Zbl

[27] G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms. II. Quantum Birkhoff normal forms, Ann. Henri Poincaré 1(2000), 249–279. | MR | Zbl

[28] J. Rauch and M. Taylor, Decay of solutions to nondissipative hyperbolic systems on compact manifolds, Comm. Pure Appl. Math. 28 (1975), 501–523. | MR | Zbl

[29] J. Sjöstrand, Singularités analytiques microlocales, Astérisque, 1982. | Numdam | MR | Zbl

[30] J. Sjöstrand, Function space associated to global $I$–Lagrangian manifolds, Structure of solutions of differential equations (Katata/Kyoto, 1995), 369–423, World Sci. Publishing, River Edge, NJ, 1996. | MR | Zbl

[31] J. Sjöstrand, Asymptotic distribution of eigenfrequencies for damped wave equations, Publ. Res. Inst. Math. Sci. 36 (2000), 573–611. | MR | Zbl

[32] J. Sjöstrand, Perturbations of selfadjoint operators with periodic classical flow, RIMS Kokyuroku 1315 (April 2003), “Wave Phenomena and asymptotic analysis”, 1–23.

[33] J. Sjöstrand and M. Zworski, Asymptotic distribution of resonances for convex obstacles, Acta Math. 183 (1999), 191–253. | MR | Zbl

[34] A. Weinstein, Asymptotics of eigenvalue clusters for the Laplacian plus a potential, Duke Math. J. 44 (1977), 883–892. | MR | Zbl