Vũ Ngọc, San
The Quantum Birkhoff Normal Form and Spectral Asymptotics
Journées équations aux dérivées partielles, Tome (2006) , p. 1-12
doi : 10.5802/jedp.37
URL stable : http://www.numdam.org/item?id=JEDP_2006____A10_0

Classification:  58J40,  58J50,  58K50,  47B35,  53D20,  81S10
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,), 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.

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