15 June 2008 Spectral asymptotics via the semiclassical Birkhoff normal form
Laurent Charles, San Vũ Ngọc
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Duke Math. J. 143(3): 463-511 (15 June 2008). DOI: 10.1215/00127094-2008-026


This article gives a simple treatment of the quantum Birkhoff normal form for semiclassical pseudodifferential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised nondegenerate 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,ħ) and gives improvements and new proofs for many of the results in the field. In the completely resonant case, we show that the pseudodifferential 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. In the case of polynomial differential operators, a combinatorial trace formula is obtained


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Laurent Charles. San Vũ Ngọc. "Spectral asymptotics via the semiclassical Birkhoff normal form." Duke Math. J. 143 (3) 463 - 511, 15 June 2008. https://doi.org/10.1215/00127094-2008-026


Published: 15 June 2008
First available in Project Euclid: 3 June 2008

zbMATH: 1154.58015
MathSciNet: MR2423760
Digital Object Identifier: 10.1215/00127094-2008-026

Primary: 58J50
Secondary: 47B35 , 53D20 , 58J40 , 58K50 , 81S10

Rights: Copyright © 2008 Duke University Press


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Vol.143 • No. 3 • 15 June 2008
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