AI 中文总结
本文扩展了有理多边形本征函数的均匀分布结果至非对角元，还给出2-环面相同结果的另一证明，依托方向台球流的弱混合性开展研究。
AI 中文摘要
施尼雷尔曼-泽尔迪奇-科兰·德·韦尔迪耶定理及其弱混合扩展将量子遍历性与量子混合和测地流的遍历性及弱混合关联起来；平环面等可积系统通常不满足这些性质。针对位置依赖可观测量，马克洛夫与鲁德尼克证明了有理多边形几乎所有本征函数的均匀分布性；本文将该结果扩展至不包含环面的一类有理多边形的非对角元，利用几乎所有方向的方向台球流的弱混合性，还给出了2-环面相同结果的另一证明。
英文摘要
The Shnirelman-Zelditch-Colin de Verdière theorem and its weak mixing extension relate quantum ergodicity and quantum mixing with ergodicity and weak mixing of the geodesic flow. Integrable systems, such as the flat torus, do not satisfy either in general. Restricting to position-dependent observables, Marklof and Rudnick established equidistribution for almost all eigenfunctions of rational polygons. In this note, we extend this result to off-diagonal elements for a subset of rational polygons not including the torus using weak mixing of the directional billiard flow for almost all directions. Additionally, we provide a different proof that establishes the same result for $2$-tori.
