Spectral asymptotics via the semiclassical Birkhoff normal form

Laurent Charles (1) , San Vu Ngoc (2, 3)

1 IMJ - Institut de Mathématiques de Jussieu
2 IF - Institut Fourier
3 IRMAR - Institut de Recherche Mathématique de Rennes

This article gives 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 $(E,\h)$, 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. In the case of polynomial differential operators, a combinatorial trace formula is obtained.

Mots clés: Birkhoff normal form; resonances; pseudo-differential operators; spectral asymptotics; symplectic reduction; Toeplitz operators; eigenvalue cluster

Domaines: Théorie spectrale [math.SP]; Géométrie symplectique [math.SG]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Laurent Charles, San Vu Ngoc. Spectral asymptotics via the semiclassical Birkhoff normal form. Duke Mathematical Journal, 2008, 143 (3), pp.463--511. ⟨hal-00023675⟩