A Uniform Quantum Version of the Cherry Theorem

Abstract

Consider in $L^2(\mathbb{R}^2)$ the operator family $H(\epsilon):=P_0(\hbar,\omega)+\epsilon F_0$ . P ₀ is the quantum harmonic oscillator with diophantine frequency vector ω, F ₀ a bounded pseudodifferential operator with symbol decreasing to zero at infinity in phase space, and $\epsilon \in {\mathbb{C}}$ . Then there exist $\epsilon^\ast > 0$ independent of $\hbar$ and an open set $\Gamma\subset{\mathbb{C}}^2\setminus{\mathbb{R}}^2$ such that if $|\epsilon| < \epsilon^\ast$ and $\omega\in\Gamma$ , 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.