发表机构
Université Paris-Saclay（巴黎-萨克雷大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对半经典伪微分算子受高频振荡符号扰动的情形，在双曲能量面上证明了长达$|\log h|$阶时间的Gutzwiller迹公式，并揭示扰动仅改变相位。
AI 中文摘要
设$P_h$是闭黎曼流形$(M,g)$上的半经典伪微分算子，其主符号为$p$。我们考虑扰动$P_h+h^\alpha Q$，其中$0<\alpha\le 1$，$Q$是符号在$h^\beta$尺度（$2\beta<\alpha$）振荡的半经典伪微分算子。假设未扰动系统在正则能量面$p^{-1}(E)$上的经典动力学是双曲的，我们证明了扰动算子的Gutzwiller迹公式在$|\log h|$阶时间内成立。该公式将量子系统在能量$E$附近的能级分布与其经典对应物的周期轨道联系起来。扰动$h^\alpha Q$仅影响迹公式中出现的相位。
英文摘要
Let $P_h$ be a semiclassical pseudodifferential operator on a closed Riemannian manifold $(M,g)$, with principal symbol $p$. We consider perturbations $P_h+h^αQ$, with $0<α\le 1$, where $Q$ is a semiclassical pseudodifferential operator whose symbol oscillates at scale $h^β$, with $2β<α$. Assuming that the classical dynamics of the unperturbed system on a regular energy surface $p^{-1}(E)$ is hyperbolic, we prove a Gutzwiller trace formula for the perturbed operator that holds up to times of order $|\log h|$. This formula relates the distribution of energy levels of the quantum system near the energy $E$ to the periodic orbits of its classical counterpart. The perturbation $h^αQ$ affects only the phases appearing in the trace formula.
Comments110 pages, comments welcome
