A uniform quantum version of the Cherry theorem

Abstract

Consider in $L^2({\mathbb{R}}^2)$ the operator family $H(\epsilon ):=P_0(\hslash ,\omega )+\epsilon F_0$. $P_0$ 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 $\hslash$ and an open set $Ω\subset\C^2\setminus\R^2$ such that if $|\ep|<\ep^\ast$ and $\om\in\Om$ the quantum normal form near $P_0$ converges uniformly with respect to $\hslash$. This yields an exact quantization formula for the eigenvalues, and for $\hslash =0$ the classical Cherry theorem on convergence of Birkhoff's normal form for complex frequencies is recovered.

17 pages

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