Duke Mathematical Journal

Spectral asymptotics via the semiclassical Birkhoff normal form

Laurent Charles and San Vũ Ngọc

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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 $(E,\hstrok)$ 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

Article information

Duke Math. J. Volume 143, Number 3 (2008), 463-511.

First available: 3 June 2008

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Zentralblatt MATH identifier

Primary: 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]
Secondary: 58J40: Pseudodifferential and Fourier integral operators on manifolds [See also 35Sxx] 58K50: Normal forms 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15] 53D20: Momentum maps; symplectic reduction 81S10: Geometry and quantization, symplectic methods [See also 53D50]


Charles, Laurent; Vũ Ngọc, San. Spectral asymptotics via the semiclassical Birkhoff normal form. Duke Mathematical Journal 143 (2008), no. 3, 463--511. doi:10.1215/00127094-2008-026. http://projecteuclid.org/euclid.dmj/1212500464.

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