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

TL;DR

The paper addresses the semiclassical quantum problem near invariant KAM tori by constructing a Gevrey quantum Birkhoff normal form for the operator $P_h(t)$ under a weak arithmetic regime given by the $\sigma$-Bruno-Rüssmann condition. It develops a Gevrey-holomorphic pseudodifferential framework, then builds a unitary microlocal conjugation $U_h(t)$ through two quantized transformations $T_{1h}$ and $T_{2h}$, followed by a normal-form conjugation that yields $P_h^0(t)$ with symbol $p^0(\varphi,I;t,h)=K^0(I;t,h)+R^0(\varphi,I;t,h)$, where $R^0$ vanishes to infinite order on the Cantor non-resonant set $E_{\kappa}(t)$. The analysis hinges on solving a Gevrey-cohomological (homological) equation on a parameter-dependent Cantor set, leveraging the subexponential Fourier decay dictated by Gevrey regularity to tame small divisors without Fourier truncation. The results extend quantum normal-form methods to Gevrey symbols under weak non-resonance, providing a robust foundation for spectral and microlocal study near KAM tori and uniform-in-$t$ estimates. This advances semiclassical analysis by linking Gevrey regularity, weakened arithmetic conditions, and quantum normal forms for invariant-torus dynamics.

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 $Λ$ 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 $σ>1$.

Theorems & Definitions (7)

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