KAM Tori and Quantum Birkhoff Normal Forms
Séminaire Équations aux dérivées partielles (Polytechnique) (1999-2000), Talk no. 19, 13 p.

This talk is concerned with the Kolmogorov-Arnold-Moser (KAM) theorem in Gevrey classes for analytic hamiltonians, the effective stability around the corresponding KAM tori, and the semi-classical asymptotics for Schrödinger operators with exponentially small error terms. Given a real analytic Hamiltonian H close to a completely integrable one and a suitable Cantor set Θ defined by a Diophantine condition, we find a family Λ ω ,ωΘ, of KAM invariant tori of H with frequencies ωΘ which is Gevrey smooth with respect to ω in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union Λ of the KAM tori which can be viewed as a Birkhoff normal form (BNF) of H around Λ. This leads to effective stability of the quasiperiodic motion near Λ. We investigate the semi-classical asymptotics of a Schrödinger type operator with a principal symbol H. We obtain semiclassical quasimodes with exponentially small error terms which are associated with the Gevrey family of KAM tori Λ ω ,ωΘ. To do this we construct a quantum Birkhoff normal form (QBNF) of the Schrödinger operator around Λ in suitable Gevrey classes starting from the BNF of H. As an application, we obtain a sharp lower bound for the counting function of the resonances which are exponentially close to a suitable compact subinterval of the real axis.

@article{SEDP_1999-2000____A19_0,
     author = {Popov, Georgi},
     title = {KAM Tori and Quantum Birkhoff Normal Forms},
     journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique)},
     publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
     year = {1999-2000},
     note = {talk:19},
     mrnumber = {1813182},
     zbl = {1056.37078},
     language = {en},
     url = {http://www.numdam.org/item/SEDP_1999-2000____A19_0}
}
Popov, Georgi. KAM Tori and Quantum Birkhoff Normal Forms. Séminaire Équations aux dérivées partielles (Polytechnique) (1999-2000), Talk no. 19, 13 p. http://www.numdam.org/item/SEDP_1999-2000____A19_0/

[1] N. Burq, Absence de résonance près du réel pour l’opérateur de Schrödinger, Seminair de l’Equations aux Dérivées Partielles, n o 17, Ecole Polytechnique, 1997/1998 | Numdam | Zbl 02124190

[2] F. Cardoso, G. Popov, Quasimodes with exponentially small errors associated with broken elliptic rays, in preparation | Zbl 01908008

[3] Y. Colin de Verdière, Quasimodes sur les variétés Riemanniennes, Inventiones Math., Vol. 43, 1977, pp. 15-52 | MR 501196 | Zbl 0449.53040

[4] V. Lazutkin, KAM theory and semiclassical approximations to eigenfunctions, Springer-Verlag, Berlin, 1993. | MR 1239173 | Zbl 0814.58001

[5] P. Lochak, Canonical perturbation theory:an approach based on joint approximations, Uspekhi Mat. Nauk, Vol. 47, 6, 1992, pp. 59-140 (in Russian); translation in: Russian Math. Surveys, Vol. 47, 6, 1992, pp. 57-133. | MR 1209145 | Zbl 0795.58042

[6] G. Popov, Invariant tori effective stability and quasimodes with exponentially small error term I - Birkhoff normal forms, Ann. Henri Poincaré, 2000, to appear. | MR 1770799 | Zbl 0970.37050

[7] G. Popov, Invariant tori effective stability and quasimodes with exponentially small error term II - Quantum Birkhoff normal forms, Ann. Henri Poincaré, 2000, to appear. | MR 1770800 | Zbl 1002.37028

[8] J. Pöschel, Lecture on the classical KAM Theorem, School on dynamical systems, May 1992, International center for science and high technology, Trieste, Italy

[9] J. Pöschel, Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z., Vol. 213, 1993, pp. 187-217. | MR 1221713 | Zbl 0857.70009

[10] J. Sjöstrand, A trace formula and review of some estimates for resonances. In: L. Rodino (eds.) Microlocal analysis and spectral theory. Nato ASI Series C: Mathematical and Physical Sciences, 490, pp. 377-437: Kluwer Academic Publishers 1997 | MR 1451399 | Zbl 0877.35090

[11] J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles, Journal of AMS, Vol. 4(4), 1991, pp. 729-769. | MR 1115789 | Zbl 0752.35046

[12] Stefanov P.: Quasimodes and resonances: Sharp lower bounds, Duke Math. J., 99, 1, 1999, pp. 75-92. | MR 1700740 | Zbl 0952.47013

[13] S.-H. Tang and M. Zworski, >From quasimodes to resonances, Math. Res. Lett., 5, 1998, pp. 261-272. | MR 1637824 | Zbl 0913.35101