KAM theory at the Quantum resonance
Huanhuan Yuana, Yong Li
TL;DR
The paper advances quantum KAM theory for semiclassical operators with partial frequency resonances by constructing a resonant quantum normal form that yields an explicit quantization of energy levels. Through a Gevrey-symbol, microlocal framework, it couples a classical normal-form reduction and a KAM iteration with a quantum conjugation scheme to produce an exponentially accurate remainder. The main contributions are (i) a detailed spectral formula showing quasi-periodic and resonant clustering terms, (ii) a proof of semiclassical scarring of eigenfunctions on lower-dimensional invariant tori, and (iii) robust frequency-set estimates ensuring persistence of the nonresonant structure under perturbation. The results illuminate how classical resonance geometry shapes quantum spectra and localization, with potential implications for spectral stability and quantum ergodicity in near-integrable systems.
Abstract
We consider the semiclassical operator $\hat{H}(ε,h):=H_{0}(hD_{x})+ε\tilde{P}_{0}$ on $L^{2}(\mathbb{R}^{l})$, where the symbol of $\hat{H}(ε,h)$ corresponds to a perturbed classical Hamiltonian of the form: \begin{align*} H(x,y,ε)=H_{0}(y)+εP_{0}(x,y). \end{align*} Here, $\tilde{P}_{0}=Op_{h}^{W}(P_{0})$ is a bounded pseudodifferential operator with a holomorphic symbol that decays to zero at infinity, and $ε\in \mathbb{R}$ is a small parameter. We establish that for small $|ε|<ε^{*}$, there exists a frequency $ω(ε)$ satisfying condition \eqref{b}, such that the spectrum of $\hat{H}(ε,h)$ is given by the quantization formula: \begin{align*} E(n_{y},E_{u},E_{v},ε,h)=\varepsilon(h,ε)+h\sum_{j=1}^{d}ω_{j}(n_{y}^{j}+\frac{\vartheta_{j}}{4})+\fracε{2}\bigg(\sum_{j=1}^{d_{0}}λ_{j} (n_{u}^{j}+\frac{1}{2})+\sum_{j=1}^{d_{0}}\tildeλ_{j}(n_{v}^{j}+\frac{1}{2})\bigg)+O(ε\exp(-ch^{\frac{1}{α-1}})), \end{align*} where $α>1$ is Gevrey index, $\vartheta$ represents the Maslov index of the torus. This spectral expression captures the detailed structure of the perturbed system, reflecting the influence of partial resonances in the classical dynamics. In particular, the resonance-induced quadratic terms give rise to clustering of eigenvalues, determined by the eigenvalues $λ_{j}$ and $\tildeλ_{j}$ of the associated quadratic form in the resonant variables. Moreover, the corresponding eigenfunctions exhibit semiclassical localization-quantum scarring-on lower-dimensional invariant tori formed via partial splitting under resonance.