Uniform semiclassical estimates for the propagation of quantum observables
Uniform semiclassical estimates for the propagation of quantum observables
复制标题
量子可观测量传播的均匀半经典估计
DOI：
10.1215/s0012-7094-02-11122-3
复制
发表时间：
2002
影响因子：
2.5
通讯作者：
D. Robert
中科院分区：
文献类型：
--
作者：
A. Bouzouina;D. Robert
我们在这里证明，对于 C 哈密尔顿量在无穷大处最多二次增长，量子可观测量传播的半经典渐近展开式在任何阶上都一致由参数在时间上呈线性的指数项支配。特别是，我们恢复了埃伦菲斯特时间以保证半经典近似的有效性。这扩展了[BGP]中证明的结果。此外，如果哈密顿量和初始可观测量在相空间的复邻域中是全纯的，我们证明量子可观测量是解析的半经典可观测量。还给出了关于可观测量的大时间行为的其他结果，重点是经典动力学。特别是，为经典可积系统建立了精确的 Gevrey 估计。
We prove here that the semiclassical asymptotic expansion for the propagation of quantum observables, for C -Hamiltonians growing at most quadratically at infinity, is uniformly dominated at any order by an exponential term whose argument is linear in time. In particular, we recover the Ehrenfest time for the validity of the semiclassical approximation. This extends the result proved in [ BGP]. Furthermore, if the Hamiltonian and the initial observables are holomorphic in a complex neighborhood of the phase space, we prove that the quantum observable is an analytic semiclassical observable. Other results about the large time behavior of observables with emphasis on the classical dynamic are also given. In particular, precise Gevrey estimates are established for classically integrable systems.