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A uniform quantum version of the Cherry theorem

Carlos Villegas Blas·Sandro Graffi
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Abstract

Consider in $L^2(\R^2)$ the operator family $H(ε):=P_0(\hbar,ω)+εF_0$. $P_0$ 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 $Ω\subset\C^2\setminus\R^2$ such that if $|\ep|<\ep^\ast$ and $\om\in\Om$ the quantum normal form near $P_0$ converges uniformly with respect to $\hbar$. 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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Carlos Villegas Blas, Sandro Graffi. 2007-02-08. A uniform quantum version of the Cherry theorem. https://arxiv.org/abs/math-ph/0702021

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