Quantum Geometry and Quantum Mechanics of Integrable Systems

Details Quantum Geometry and Quantum Mechanics of Integrable Systems Quantum Geometry and Quantum Mechanics of Integrable Systems

2009-08-14

arxiv.org/abs/0908.2048v1

Russian Journal of Mathematical Physics, Vol.16, No.1, 2009, pp.81-92

Mathematics

Symplectic Geometry

Latex, 16p

Scientific paper

10.1134/S1061920809010051

Quantum integrable systems and their classical counterparts are considered. We show that the symplectic structure and invariant tori of the classical system can be deformed by a quantization parameter $\hbar$ to produce a new (classical) integrable system. The new tori selected by the $\hbar$-equidistance rule represent the spectrum of the quantum system up to $O(\hbar^\infty)$ and are invariant under quantum dynamics in the long-time range $O(\hbar^{-\infty})$. The quantum diffusion over the deformed tori is described. The analytic apparatus uses quantum action-angle coordinates explicitly constructed by an $\hbar$-deformation of the classical action-angles.

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