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
@incollection{JEDP_2006____A10_0,
author = {V\~{u} Ngọc, San},
title = {The {Quantum} {Birkhoff} {Normal} {Form} and {Spectral} {Asymptotics}},
booktitle = {},
series = {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/
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