CONVERGENCE OF A QUANTUM NORMAL FORM AND AN EXACT QUANTIZATION FORMULA

Let the quantization of the linear flow of diophantine frequencies $\om$ over the torus $\T^l$, $l>1$, namely the Schrödinger operator $-i\hbar\omega\cdot\nabla$ on $L^2(\T^l)$, be perturbed by the quantization of a function $\V_\om: \R^l\times\T^l\to\R$ of the form \vskip 5pt\noindent $$ \V_\om(\xi,x)=\V(z\circ \L_\om(\xi),x),\quad \L_\om(\xi):= \om_1\xi_1+\ldots+\om_l\xi_l $$ \vskip 4pt\noindent where $z\mapsto \V(z,x): \R\times\T^l \to\R$ is real-holomorphic. We prove that the corresponding quantum normal form converges uniformly with respect to $\hbar\in [0,1]$. Since the quantum normal form reduces to the classical one for $\hbar=0$, this result simultaneously yields an exact quantization formula for the quantum spectrum, as well as a convergence criterion for the Birkhoff normal form, valid for a class of perturbations holomorphic away from the origin. The main technical aspect concerns the quantum homological equation $\ds {[F(-i\hbar\om\cdot\nabla),W]}/{i\hbar}+V=N$, $F:\R\to\R$ being a smooth function $\ep-$close to the identity. Its solution is constructed, and estimated uniformly with respect to $\hbar\in [0,1]$, by solving the equation $\{F(\L_\om),\W\}_M+\V=\N$ for the corresponding symbols. Here $\{\cdot,\cdot\}_M$ stands for the Moyal bracket. As a consequence, the KAM iteration for the symbols of the quantum operators can be implemented, and its convergence proved, uniformly with respect to $(\xi,\hbar,\ep)\in \R^l\times [0,1]\times \{\ep\in\C\,|\;|\ep|0$ is explicitly estimated in terms only of the diophantine constants. This in turn entails the uniform convergence of the quantum normal form