  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.
