标题： 大时间的半经典波传播 显示英文标题

标题： Semiclassical wave propagation for large times

摘要： 我们研究在黎曼流形上随时间变化的薛定谔方程的解，初始条件由拉格朗日态给出，这些初始条件具有振荡性质。半经典近似描述了当h较小时（h为半经典参数）这些解的情况，但它们在长时间范围内的准确性通常仅被理解到埃伦费斯特定时T ~ ln(1/h)，最困难的情况是底层经典系统是混沌的。我们证明，在常负曲率曲面上，当拉格朗日态与测地流的不稳定流形相关时，半经典近似在时间至少达到T ~ h^(-1/2)时仍然准确。 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.