Localization of Quantum States and Landscape Functions
Localization of Quantum States and Landscape Functions
复制标题
量子态和景观函数的局域化
DOI：
10.1090/proc/13343
复制
发表时间：
2015
期刊：
影响因子：
--
通讯作者：
S. Steinerberger
中科院分区：
文献类型：
--
作者：
S. Steinerberger
非均匀介质中的本征函数具有很强的局域化特性。 Filoche \& Mayboroda 表明，求解 $(-\Delta + V)u = 1$ 的函数 $u$ 通过不等式 $$|\phi(x)| 控制特征函数 $(-\Delta + V)\phi = \lambda\phi$ 的行为\leq \lambda u(x) \|\phi\|_{L^{\infty}}.$$ 这种不等式已被证明在预测定位方面非常有效，最近 Arnold、David、Jerison、Mayboroda \& Filoche 将 $1/u$ 连接到本征函数的衰减特性。我们的目标是阐明景观的属性：主要成分是通过将 $\phi(x)$ 写为从 $x$ 开始的布朗运动 $\omega(\cdot)$ 的平均值而获得的局部变化估计 $$\phi(x) = \mathbb{E}_{x}\left(\phi(\omega(t)) e^{\lambda t-\int_{0}^{t}{V(\omega(z))dz}} \right)。$$这种变化估计将保证 $\phi$ 在一个小球中至少变化 2 倍，这隐含地创建了一个景观，其与我们讨论的 $1/u$ 的关系。
Eigenfunctions in inhomogeneous media can have strong localization properties. Filoche \& Mayboroda showed that the function $u$ solving $(-\Delta + V)u = 1$ controls the behavior of eigenfunctions $(-\Delta + V)\phi = \lambda\phi$ via the inequality $$|\phi(x)| \leq \lambda u(x) \|\phi\|_{L^{\infty}}.$$ This inequality has proven to be remarkably effective in predicting localization and recently Arnold, David, Jerison, Mayboroda \& Filoche connected $1/u$ to decay properties of eigenfunctions. We aim to clarify properties of the landscape: the main ingredient is a localized variation estimate obtained from writing $\phi(x)$ as an average over Brownian motion $\omega(\cdot)$ in started in $x$ $$\phi(x) = \mathbb{E}_{x}\left(\phi(\omega(t)) e^{\lambda t-\int_{0}^{t}{V(\omega(z))dz}} \right).$$ This variation estimate will guarantee that $\phi$ has to change at least by a factor of 2 in a small ball, which implicitly creates a landscape whose relationship with $1/u$ we discuss.