The Quantum Birkhoff Normal Form and Spectral Asymptotics
Journées équations aux dérivées partielles (2006), article no. 10, 12 p.

In this talk we explain a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate potential well, yielding uniform estimates in the energy E. This permits a detailed study of the spectrum in various asymptotic regions of the parameters (E,), and gives improvements and new proofs for many of the results in the field. In the completely resonant case we show that the pseudo-differential operator can be reduced to a Toeplitz operator on a reduced symplectic orbifold. Using this quantum reduction, new spectral asymptotics concerning the fine structure of eigenvalue clusters are proved.

DOI: 10.5802/jedp.37
Classification: 58J40, 58J50, 58K50, 47B35, 53D20, 81S10
Keywords: Birkhoff normal form, resonances, pseudo-differential operators, spectral asymptotics, symplectic reduction, Toeplitz operators, eigenvalue cluster
Vũ Ngọc, San 1

1 Institut Fourier (UMR 5582), Université Joseph Fourier, Grenoble 1, BP 74, 38402-Saint Martin d’Hères Cedex, France.
@article{JEDP_2006____A10_0,
     author = {V\~{u} Ngọc, San},
     title = {The {Quantum} {Birkhoff} {Normal} {Form} and {Spectral} {Asymptotics}},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {10},
     pages = {1--12},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2006},
     doi = {10.5802/jedp.37},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.37/}
}
TY  - JOUR
AU  - Vũ Ngọc, San
TI  - The Quantum Birkhoff Normal Form and Spectral Asymptotics
JO  - Journées équations aux dérivées partielles
PY  - 2006
SP  - 1
EP  - 12
PB  - Groupement de recherche 2434 du CNRS
UR  - http://www.numdam.org/articles/10.5802/jedp.37/
DO  - 10.5802/jedp.37
LA  - en
ID  - JEDP_2006____A10_0
ER  - 
%0 Journal Article
%A Vũ Ngọc, San
%T The Quantum Birkhoff Normal Form and Spectral Asymptotics
%J Journées équations aux dérivées partielles
%D 2006
%P 1-12
%I Groupement de recherche 2434 du CNRS
%U http://www.numdam.org/articles/10.5802/jedp.37/
%R 10.5802/jedp.37
%G en
%F JEDP_2006____A10_0
Vũ Ngọc, San. The Quantum Birkhoff Normal Form and Spectral Asymptotics. Journées équations aux dérivées partielles (2006), article  no. 10, 12 p. doi : 10.5802/jedp.37. http://www.numdam.org/articles/10.5802/jedp.37/

[1] G.D. Birkhoff. Dynamical systems. AMS, 1927. | Zbl

[2] L. Charles. Toeplitz operators and Hamiltonian torus actions. Jour. Funct. Analysis, 2006. To appear. | MR | Zbl

[3] L. Charles and S. Vũ Ngọc. Spectral asymptotics via the semiclassical birkhoff normal form. math.SP/0605096.

[4] Y. Colin de Verdière. Sur le spectre des opérateurs elliptiques à bicaractéristiques toutes périodiques. Comment. Math. Helv., 54:508–522, 1979. | MR | Zbl

[5] J. J. Duistermaat. Non-integrability of the 1:1:2 resonance. Ergodic Theory Dynamical Systems, 4:553–568, 1984. | MR | Zbl

[6] B. Helffer and J. Sjöstrand. Multiple wells in the semi-classical limit. I. Comm. Partial Differential Equations, 9:337–408, 1984. | MR | Zbl

[7] M. Hitrik, J. Sjöstrand, and S. Vũ Ngọc. Diophantine tori and spectral asymptotics for non-selfadjoint operators. math.SP/0502032. À paraître dans Am. J. Math., 2005.

[8] R. Pérez-Marco. Convergence or generic divergence of the Birkhoff normal form. Ann. of Math. (2), 157(2):557–574, 2003. | MR | Zbl

[9] G. Popov. Invariant tori, effective stability, and quasimodes with exponentially small error terms. II. Quantum Birkhoff normal forms. Ann. Henri Poincaré, 1(2):249–279, 2000. | MR | Zbl

[10] B. Simon. Semiclassical analysis of low lying eigenvalues I. Ann. Inst. H. Poincaré. Phys. Théor., 38(3):295–307, 1983. a correction in 40:224. | Numdam | MR | Zbl

[11] J. Sjöstrand. Semi-excited states in nondegenerate potential wells. Asymptotic Analysis, 6:29–43, 1992. | MR | Zbl

[12] S. Vũ Ngọc. Sur le spectre des systèmes complètement intégrables semi-classiques avec singularités. PhD thesis, Université Grenoble 1, 1998.

[13] A. Weinstein. Asymptotics of eigenvalue clusters for the laplacian plus a potential. Duke Math. J., 44(4):883–892, 1977. | MR | Zbl

[14] Nguyên Tiên Zung. Convergence versus integrability in Birkhoff normal form. Ann. of Math. (2), 161(1):141–156, 2005. | MR | Zbl

Cited by Sources: