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A Uniform Quantum Version of the Cherry Theorem

Authors:	Sandro Graffi Carlos Villegas-Blas

Institution:	(1) Dipartimento di Matematica, Università di Bologna, Bologna, Italy;(2) Instituto de Matematicas, Universitad Nacional Autonoma de Mexico, Unidad Cuernvaca, Cuernavaca, Mexico

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. Partially supported by PAPIIT-UNAM IN106106-2.

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