A quantum Cherry theorem for perturbations of the plane rotator

{We consider on $L^2(\T^2)$ the \Sc\ operator family $L_\ep: \ep\in\R$ with domain and action defined as $\ds D(L_\ep)=H^2(\T^2),\; L_\ep u=-\frac12\hbar^2(\alpha_1\partial_{\phi_1}^2+\alpha_2\partial_{\phi_2}^2)u-i\hbar(\gamma_1\partial_{\phi_1}+\gamma_2\partial_{\phi_2})u+\ep Vu. $ Here $\ep\in\R$, $\alpha=(\alpha_1,\alpha_2)$, $\gamma=(\gamma_1,\gamma_2)$ are vectors of complex non-real frequencies, and $V$ a pseudodifferential operator of order zero. $L_\ep$ represents the Weyl quantization of the Hamiltonian family $ \L_\ep(\xi,x)=\frac12(\alpha_1\xi_1^2+\alpha_2\xi_2^2)+\gamma_1\xi_1+\gamma_2\xi_2+\ep \V(\xi,x)$ defined on the phase space $\R^2\times\T^2$, where $\V(\xi,x)\in C^2(\R^2\times \T^2;\R)$. We prove the uniform convergence with respect to $\hbar\in [0,1]$ of the quantum normal form, which reduces to the classical one for $\hbar=0$. This result simultaneously entails an exact quantization formula for the quantum spectrum as well as a convergence criterion for the classical Birkhoff normal form generalizing a well known theorem of Cherry to a class of perturbations of a non self-adoint quantum plane rotator.}