Quantum Geometry and Quantum Mechanics of Integrable Systems

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.