Quantum Birkhoff Normal Form in the $σ$-Bruno-Rüssmann non-resonant condition

The aim of this paper is to construct a Gevrey quantum Birkhoff normal form for the $h$-differential operator $P_{h}(t),$ where $t\in(-\frac{1}{2},\frac{1}{2})$, in the neighborhood of the union $Λ$ of KAM tori. This construction commences from an appropriate Birkhoff normal form of $H$ around $Λ$ and proceeds under the $σ$-Bruno-Rüssmann condition with $σ>1$.

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