论文标题
通过近似半群属性扩散的量子扩散
Quantum Diffusion via an Approximate Semigroup Property
论文作者
论文摘要
在本文中，我们引入了一种新方法，以实现L. Erdos，M。Salmhofer和H.T.首次研究的弱随机Schrodinger方程的扩散极限。 Yau。我们的方法是基于演化算子的波袋分解，这使我们能够将Duhamel系列解释为分段线性路径上不可或缺的组成部分。我们将这些路径的几何形状与图解扩展的组合特征相结合，这使我们能够将误差项表示为在某种程度上是非凡的路径上的积分。这些错误项是使用几何参数界定的。然后，主要术语被证明具有半群属性，这使我们可以迭代地增加有效扩散的有效性时间。这是从随机的schrodinger方程中的有效扩散方程的第一个推导，该方程在尺寸$ d \ geq 2 $中有效。
In this paper we introduce a new approach to the diffusive limit of the weakly random Schrodinger equation, first studied by L. Erdos, M. Salmhofer, and H.T. Yau. Our approach is based on a wavepacket decomposition of the evolution operator, which allows us to interpret the Duhamel series as an integral over piecewise linear paths. We relate the geometry of these paths to combinatorial features of a diagrammatic expansion which allows us to express the error terms in the expansion as an integral over paths that are exceptional in some way. These error terms are bounded using geometric arguments. The main term is then shown to have a semigroup property, which allows us to iteratively increase the timescale of validity of an effective diffusion. This is the first derivation of an effective diffusion equation from the random Schrodinger equation that is valid in dimensions $d\geq 2$.
