标题： 大时间尺度下的半经典波传播 显示英文标题

标题： Semiclassical wave propagation for large times

摘要： 我们研究了定义在黎曼流形上带有震荡初始条件的薛定谔方程的时间相关解，这些初始条件由拉格朗日态给出。 半经典近似在小参数 h（其中 h 是半经典的参数）下描述了这些解，但对于大时间的精确度，通常只能理解到 Ehrenfest 时间尺度 T ~ ln(1/h)，并且最困难的情况是当底层的经典系统是混沌的时候。 我们证明，在曲率为常数负值的曲面上，如果拉格朗日态与测地线流的不稳定流形相关联，则在至少直到 T ~ h^(-1/2) 的时间内，半经典近似仍然准确。 显示英文摘要

摘要： We study solutions of the time dependent Schr\"odinger equation on Riemannian manifolds with oscillatory initial conditions given by Lagrangian states. Semiclassical approximations describe these solutions for small h (where h is the semiclassical parameter), but their accuracy for large times is in general only understood up to the Ehrenfest time T ~ ln(1/h), and the most difficult case is the one where the underlying classical system is chaotic. We show that on surfaces of constant negative curvature semiclassical approximations remain accurate for times at least up to T ~ h^(-1/2) in the case that the Lagrangian state is associated with an unstable manifold of the geodesic flow.