Quantum Birkhoff Normal Form in the $\sigma$-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 $\mathrm{\Lambda }$ of KAM tori. This construction commences from an appropriate Birkhoff normal form of $H$ around $\mathrm{\Lambda }$ and proceeds under the $\sigma$-Bruno-Rüssmann condition with $\sigma >1$.