The Quantum Birkhoff Normal Form and Spectral Asymptotics

Vũ Ngọc, San. "The Quantum Birkhoff Normal Form and Spectral Asymptotics." Journées Équations aux dérivées partielles (2006): 1-12. <http://eudml.org/doc/10616>.

@article{VũNgọc2006,
abstract = {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 $(E,\hbar)$, 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.},
affiliation = {Institut Fourier (UMR 5582), Université Joseph Fourier, Grenoble 1, BP 74, 38402-Saint Martin d’Hères Cedex, France.},
author = {Vũ Ngọc, San},
journal = {Journées Équations aux dérivées partielles},
keywords = {Birkhoff normal form; resonances; pseudo-differential operators; spectral asymptotics; symplectic reduction; Toeplitz operators; eigenvalue cluster},
language = {eng},
month = {6},
pages = {1-12},
publisher = {Groupement de recherche 2434 du CNRS},
title = {The Quantum Birkhoff Normal Form and Spectral Asymptotics},
url = {http://eudml.org/doc/10616},
year = {2006},
}

TY - JOUR
AU - Vũ Ngọc, San
TI - The Quantum Birkhoff Normal Form and Spectral Asymptotics
JO - Journées Équations aux dérivées partielles
DA - 2006/6//
PB - Groupement de recherche 2434 du CNRS
SP - 1
EP - 12
AB - 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 $(E,\hbar)$, 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.
LA - eng
KW - Birkhoff normal form; resonances; pseudo-differential operators; spectral asymptotics; symplectic reduction; Toeplitz operators; eigenvalue cluster
UR - http://eudml.org/doc/10616
ER -