A local quantum version of the Kolmogorov theorem

Consider in $L^2 (R^l)$ the operator family $H(epsilon):=P_0(hbar,omega)+epsilon Q_0$. $P_0$ is the quantum harmonic oscillator with diophantine frequency vector $om$, $Q_0$ a bounded pseudodifferential operator with symbol holomorphic and decreasing to zero at infinity, and $epinR$. Then there exists $ep^ast >0$ with the property that if $|ep|<ep^ast$ there is a diophantine frequency $om(ep)$ such that all eigenvalues $E_n(hbar,ep)$ of $H(ep)$ near 0 are given by the quantization formula $E_alpha(hbar,ep)= {cal E}(hbar,ep)+laom(ep),alpharahbar +|om(ep)|hbar/2 + ep O(alphahbar)^2$, where $alpha$ is an $l$-multi-index.