A uniform quantum version of the Cherry theorem

Consider in $L^2(\R^2)$ the operator family $H(\epsilon):=P_0(\hbar,\omega)+\epsilon F_0$. $P_0$ is the quantum harmonic oscillator with diophantine frequency vector \om, $F_0$ a bounded pseudodifferential operator with symbol decreasing to zero at infinity in phase space, and \ep\in\C. Then there exist \ep^\ast >0 independent of $\hbar$ and an open set \Omega\subset\C^2\setminus\R^2 such that if |\ep|<\ep^\ast and \om\in\Om the quantum normal form near $P_0$ converges uniformly with respect to $\hbar$. This yields an exact quantization formula for the eigenvalues, and for $\hbar=0$ the classical Cherry theorem on convergence of Birkhoff's normal form for complex frequencies is recovered.Comment: 17 page