 0733 Sandro Graffi, Carlos Villegas Blas
 A UNIFORM QUANTUM VERSION OF THE CHERRY THEOREM
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Feb 8, 07

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Abstract. 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.
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