KAM Tori and Quantum Birkhoff Normal Forms

Georgi Popov[1]

  • [1] Département de Mathématiques, UMR 6629, Université de Nantes - CNRS, B.P. 92208, 44322 Nantes-Cedex 03, France

Séminaire Équations aux dérivées partielles (1999-2000)

  • Volume: 1999-2000, page 1-13

Abstract

top
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 , ω ∈ Θ , of KAM invariant tori of H with frequencies ω ∈ Θ 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 , ω ∈ Θ . 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.

How to cite

top

Popov, Georgi. "KAM Tori and Quantum Birkhoff Normal Forms." Séminaire Équations aux dérivées partielles 1999-2000 (1999-2000): 1-13. <http://eudml.org/doc/10993>.

@article{Popov1999-2000,
abstract = {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 $\Theta $ defined by a Diophantine condition, we find a family $\Lambda _\omega ,\ \omega \in \Theta $, of KAM invariant tori of $H$ with frequencies $\omega \in \Theta $ which is Gevrey smooth with respect to $\omega $ in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union $\Lambda $ of the KAM tori which can be viewed as a Birkhoff normal form (BNF) of $H$ around $\Lambda $. This leads to effective stability of the quasiperiodic motion near $\Lambda $. 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 $\Lambda _\omega ,\ \omega \in \Theta $. To do this we construct a quantum Birkhoff normal form (QBNF) of the Schrödinger operator around $\Lambda $ 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.},
affiliation = {Département de Mathématiques, UMR 6629, Université de Nantes - CNRS, B.P. 92208, 44322 Nantes-Cedex 03, France},
author = {Popov, Georgi},
journal = {Séminaire Équations aux dérivées partielles},
keywords = {Kolmogorov-Arnold-Moser theorem; Gevrey classes; analytic Hamiltonians; effective stability; semiclassical asymptotics; Schrödinger operators; quasiperiodic motion },
language = {eng},
pages = {1-13},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {KAM Tori and Quantum Birkhoff Normal Forms},
url = {http://eudml.org/doc/10993},
volume = {1999-2000},
year = {1999-2000},
}

TY - JOUR
AU - Popov, Georgi
TI - KAM Tori and Quantum Birkhoff Normal Forms
JO - Séminaire Équations aux dérivées partielles
PY - 1999-2000
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 1999-2000
SP - 1
EP - 13
AB - 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 $\Theta $ defined by a Diophantine condition, we find a family $\Lambda _\omega ,\ \omega \in \Theta $, of KAM invariant tori of $H$ with frequencies $\omega \in \Theta $ which is Gevrey smooth with respect to $\omega $ in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union $\Lambda $ of the KAM tori which can be viewed as a Birkhoff normal form (BNF) of $H$ around $\Lambda $. This leads to effective stability of the quasiperiodic motion near $\Lambda $. 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 $\Lambda _\omega ,\ \omega \in \Theta $. To do this we construct a quantum Birkhoff normal form (QBNF) of the Schrödinger operator around $\Lambda $ 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.
LA - eng
KW - Kolmogorov-Arnold-Moser theorem; Gevrey classes; analytic Hamiltonians; effective stability; semiclassical asymptotics; Schrödinger operators; quasiperiodic motion
UR - http://eudml.org/doc/10993
ER -

References

top
  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 Zbl1255.35085
  2. F. Cardoso, G. Popov, Quasimodes with exponentially small errors associated with broken elliptic rays, in preparation Zbl1136.35426
  3. Y. Colin de Verdière, Quasimodes sur les variétés Riemanniennes, Inventiones Math., Vol. 43, 1977, pp. 15-52 Zbl0449.53040MR501196
  4. V. Lazutkin, KAM theory and semiclassical approximations to eigenfunctions, Springer-Verlag, Berlin, 1993. Zbl0814.58001MR1239173
  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. Zbl0795.58042MR1209145
  6. G. Popov, Invariant tori effective stability and quasimodes with exponentially small error term I - Birkhoff normal forms, Ann. Henri Poincaré, 2000, to appear. Zbl0970.37050MR1770799
  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. Zbl1002.37028MR1770800
  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. Zbl0857.70009MR1221713
  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 Zbl0877.35090MR1451399
  11. J. Sjöstrand and M. Zworski, Complex scaling and the distribution of scattering poles, Journal of AMS, Vol. 4(4), 1991, pp. 729-769. Zbl0752.35046MR1115789
  12. Stefanov P.: Quasimodes and resonances: Sharp lower bounds, Duke Math. J., 99, 1, 1999, pp. 75-92. Zbl0952.47013MR1700740
  13. S.-H. Tang and M. Zworski, &gt;From quasimodes to resonances, Math. Res. Lett., 5, 1998, pp. 261-272. Zbl0913.35101MR1637824

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.