半经典Maupertuis-Jacobi对应：稳定谱和不稳定谱
The semi-classical Maupertuis-Jacobi correspondence: stable and unstable spectra
摘要：
当${\cal H}_λ(x,ξ)$是满足一定等能非简并条件的完全可积哈密顿量$\widetilde{\cal H}$的微扰时，研究了二维哈密顿量族$(H_λ(x,ξ), {\cal H}_λ(x,ξ))$的Maupertuis-Jacobi对应的半经典性质.假设$\hat H_λ$在$E$附近只有离散谱，且能量面$\{\widetilde{\cal H}_0={\cal E}\}$被两两不相交的Lagrange环面隔开，我们证明了$\hat H_λ$在$E$附近的大部分特征值渐近退化为$h\to0$ .这特别适用于确定被困模式的岛屿，在线性理论的水波。我们还考虑了有理环面附近的准模。最后讨论了Katok球面赤道上Maupertuis-Jacobi对应的破缺。
英文摘要：
We investigate semi-classical properties of Maupertuis-Jacobi correspondence for families of 2-D Hamiltonians $(H_λ(x,ξ), {\cal H}_λ(x,ξ))$, when ${\cal H}_λ(x,ξ)$ is the perturbation of a completely integrable Hamiltonian $\widetilde{\cal H}$ veriying some isoenergetic non-degeneracy conditions. Assuming $\hat H_λ$ has only discrete spectrum near $E$, and the energy surface $\{\widetilde{\cal H}_0={\cal E}\}$ is separated by some pairwise disjoint Lagrangian tori, we show that most of eigenvalues for $\hat H_λ$ near $E$ are asymptotically degenerate as $h\to0$. This applies in particular for the determination of trapped modes by an island, in the linear theory of water-waves. We also consider quasi-modes localized near rational tori. Finally, we discuss breaking of Maupertuis-Jacobi correspondence on the equator of Katok sphere.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
