KAM theory at the Quantum resonance

We consider the semiclassical operator $\hat{H}(\epsilon,h):=H_{0}(hD_{x})+\epsilon \tilde{P}_{0}$ on $L^{2}(\mathbb{R}^{l})$, where the symbol of $\hat{H}(\epsilon,h)$ corresponds to a perturbed classical Hamiltonian of the form: \begin{align*} H(x,y,\epsilon)=H_{0}(y)+\epsilon P_{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 $\epsilon\in \mathbb{R}$ is a small parameter. We establish that for small $|\epsilon|<\epsilon^{*}$, there exists a frequency $\omega(\epsilon)$ satisfying condition \eqref{b}, such that the spectrum of $\hat{H}(\epsilon,h)$ is given by the quantization formula: \begin{align*} E(n_{y},E_{u},E_{v},\epsilon,h)=\varepsilon(h,\epsilon)+h\sum_{j=1}^{d}\omega_{j}(n_{y}^{j}+\frac{\vartheta_{j}}{4})+\frac{\epsilon}{2}\bigg(\sum_{j=1}^{d_{0}}\lambda_{j} (n_{u}^{j}+\frac{1}{2})+\sum_{j=1}^{d_{0}}\tilde{\lambda}_{j}(n_{v}^{j}+\frac{1}{2})\bigg)+O(\epsilon\exp(-ch^{\frac{1}{\alpha-1}})), \end{align*} where $\alpha>1$ 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 $\lambda_{j}$ and $\tilde{\lambda}_{j}$ 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.