 04185 D. Borthwick, S. Graffi
 A local quantum version of the
Kolmogorov theorem
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Jun 14, 04

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Abstract. 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 $\ep\in\R$. 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 $E_\alpha(\hbar,\ep)=
{\cal E}(\hbar,\ep)+\la\om(\ep),\alpha\ra\hbar +\om(\ep)\hbar/2 + \ep O(\alpha\hbar)^2$, where $\alpha$ is an $l$multiindex.
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