Abstract
LetT 0(ħ, ω)+εV be the Schrödinger operator corresponding to the classical HamiltonianH 0(ω)+εV, whereH 0(ω) is thed-dimensional harmonic oscillator with non-resonant frequencies ω=(ω1, ... , ω d ) and the potentialV(q 1, ... ,q d) is an entire function of order (d+1)−1. We prove that the algorithm of classical, canonical perturbation theory can be applied to the Schrödinger equation in the Bargmann representation. As a consequence, each term of the Rayleigh-Schrödinger series near any eigenvalue ofT 0(ħ, ω) admits a convergent expansion in powers of ħ of initial point the corresponding term of the classical Birkhoff expansion. Moreover ifV is an even polynomial, the above result and the KAM theorem show that all eigenvalues λ n (ħ, ε) ofT 0+εV such thatnħ coincides with a KAM torus are given, up to order ε∞, by a quantization formula which reduces to the Bohr-Sommerfeld one up to first order terms in ħ.
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Communicated by G. Parisi
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Graffi, S., Paul, T. The Schrödinger equation and canonical perturbation theory. Commun.Math. Phys. 108, 25–40 (1987). https://doi.org/10.1007/BF01210701
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DOI: https://doi.org/10.1007/BF01210701