Spectral asymptotics via the semiclassical Birkhoff normal form

Details Spectral asymptotics via the semiclassical Birkhoff normal form Spectral asymptotics via the semiclassical Birkhoff normal form

2006-05-03

arxiv.org/abs/math/0605096v1

Duke Mathematical Journal 143, 3 (2008) 463--511

Mathematics

Spectral Theory

44 pages, 2 figures

Scientific paper

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.

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