Quantum Birkhoff Normal Form in the $\sigma$-Bruno-R\"{u}ssmann non-resonant condition

TL;DR

This paper develops a Gevrey quantum Birkhoff normal form for certain $h$-differential operators near KAM tori under the $\sigma$-Bruno-Rüssmann non-resonance condition, advancing quantum normal form theory in non-resonant settings.

Contribution

It introduces a Gevrey quantum Birkhoff normal form construction for $h$-differential operators near KAM tori under the $\sigma$-Bruno-Rüssmann condition, extending previous normal form methods.

Findings

Constructed a Gevrey quantum Birkhoff normal form near KAM tori.

Established the normal form under the $\sigma$-Bruno-Rüssmann non-resonance condition.

Provides a framework for analyzing quantum operators in non-resonant regimes.

Abstract

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 $\Lambda$ of KAM tori. This construction commences from an appropriate Birkhoff normal form of $H$ around $\Lambda$ and proceeds under the $\sigma$-Bruno-R\"{u}ssmann condition with $\sigma>1$.