随机薛定谔方程的存在唯一性和逼近：扩散情形
Existence, uniqueness and approximation of a stochastic Schrödinger equation: the diffusive case
摘要：
量子物理学的最新发展大量使用了所谓的“量子轨迹”。在数学上，这一理论产生了“随机薛定谔方程“，即薛定谔型方程在随机微分方程形式下的扰动。但这样的方程一般不是文献中考虑的通常类型。它们在证明解的存在性和唯一性、证明方程的物理相关性方面提出了一个严重的问题。在这篇文章中，我们集中在一个特殊的情况下：扩散的情况下，一个两能级系统。证明了相应的随机薛定谔方程解的存在唯一性。我们通过证明它们是获得量子轨迹的具体物理过程的连续时间限制来物理地证明这些方程。
英文摘要：
Recent developments in quantum physics make heavy use of so-called "quantum trajectories." Mathematically, this theory gives rise to "stochastic Schrödinger equations", that is, perturbation of Schrödinger-type equations under the form of stochastic differential equations. But such equations are in general not of the usual type as considered in the literature. They pose a serious problem in terms of justifying the existence and uniqueness of a solution, justifying the physical pertinence of the equations. In this article we concentrate on a particular case: the diffusive case, for a two-level system. We prove existence and uniqueness of the associated stochastic Schrödinger equation. We physically justify the equations by proving that they are a continuous-time limit of a concrete physical procedure for obtaining a quantum trajectory.
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