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Lagrangian rings. Multiscale asymptotics of a spectrum near resonance. (English. Russian original) Zbl 0684.58034
Funct. Anal. Appl. 21, No. 1-3, 68-70 (1987); translation from Funkts. Anal. Prilozh. 21, No. 1, 78-79 (1987).
The author shows that under the appearance of a resonance the secular symbol becomes operator-valued and its eigenvalues - terms - are given by the quantization of some auxiliary Hamiltonian system from the time of Lagrange and Laplace. In this connection the problem of removing the degeneracy can be completely solved in three cases: 1. The symmetry algebra is abelian; 2. Spectral series corresponding to stationary points of the auxiliary system; 3. Large deviations of the torus from the equilibrium state arise.
Reviewer: J.H.Tian

MSC:
37A30 Ergodic theorems, spectral theory, Markov operators
53D50 Geometric quantization
81Q15 Perturbation theories for operators and differential equations in quantum theory
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