Convergence of a quantum normal form and an exact quantization formula

The operator $-i\hbar\omega\cdot\nabla$ on $L^2(\T^l)$, quantizing the linear flow of diophantine frequencies $\om=(\om_1,\ldots,\om_l)$ over $\T^l$, $l>1$, is perturbed by the quantization of a function $\V_\om(\xi,x)=\V(\om\cdot \xi,x): \R^l\times\T^l\to\R$, $z\mapsto \V(z,x): \R\times\T^l \to\R$ real-holomorphic. The corresponding quantum normal form (QNF) is proved to converge uniformly in $\hbar\in [0,1]$. This yields non-trivial examples of quantum integrable systems, an exact quantization formula for the spectrum, and a convergence criterion for the Birkhoff normal form, valid for perturbations holomorphic away from the origin. The main technical aspect concerns the quantum homological equation. Its solution is constructed, and uniformly estimated, by solving the Moyal equation for the operator symbols. The KAM iteration can thus be implemented on the symbols, and its uniform convergence proved, for $|\ep|0$ is estimated in terms only of the diophantine constants of $\om$. This entails the QNF convergence