标度极限下Anderson模型的量子扩散
Quantum diffusion for the Anderson model in the scaling limit
摘要：
考虑随机势同分布的$d\ge 3$的$\bZ^d$上的随机薛定谔方程.用$λ$表示耦合常数，用$ψ_t$表示具有初始数据$ψ_0$的溶液。空间和时间变量按$x\sim λ^{-2 -κ/2}, t \sim λ^{-2 -κ}$和$0< κ< κ_0(d)$缩放。证明了在极限$λ\to 0$下，对于任意的$L^2$初始数据，$ψ_t$的Wigner分布的期望弱收敛于空间变量$x$中的热方程的解.扩散系数由与动量$v$相关的动能唯一确定。本文的工作是对我们以前在连续统\cite{ESYI}，\cite{ESYII}中的结果的推广.由于色散关系的水平曲面的非凸性，一些费曼图的估计涉及较多。
英文摘要：
We consider random Schrödinger equations on $\bZ^d$ for $d\ge 3$ with identically distributed random potential. Denote by $λ$ the coupling constant and $ψ_t$ the solution with initial data $ψ_0$. The space and time variables scale as $x\sim λ^{-2 -κ/2}, t \sim λ^{-2 -κ}$ with $0< κ< κ_0(d)$. We prove that, in the limit $λ\to 0$, the expectation of the Wigner distribution of $ψ_t$ converges weakly to a solution of a heat equation in the space variable $x$ for arbitrary $L^2$ initial data. The diffusion coefficient is uniquely determined by the kinetic energy associated to the momentum $v$. This work is an extension to the lattice case of our previous result in the continuum \cite{ESYI}, \cite{ESYII}. Due to the non-convexity of the level surfaces of the dispersion relation, the estimates of several Feynman graphs are more involved.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
