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 . This permits a detailed study of the spectrum in various asymptotic regions of the parameters , 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.
Mots clés : Birkhoff normal form, resonances, pseudo-differential operators, spectral asymptotics, symplectic reduction, Toeplitz operators, eigenvalue cluster
@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 -
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
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