Localization of eigenfunctions via an effective potential
Localization of eigenfunctions via an effective potential
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DOI：
10.1080/03605302.2019.1626420
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发表时间：
2019-07-08
影响因子：
1.9
通讯作者：
Mayboroda, Svitlana
中科院分区：
文献类型：
--
作者：
Arnold, Douglas N.;David, Guy;Mayboroda, Svitlana
我们考虑Lipschitz域Ω上算子的本征函数的局部化，更一般地，在有边界和无边界的流形上。在早期的工作中，本论文的两位作者证明了景观的显着能力，定义为Lu = 1的解决方案，预测本地化的本征函数的位置。在这里，我们解释和证明一个新的框架，揭示了丰富的本征函数和本征值的详细画像。我们发现，景观功能的倒数，1/u，作为一个有效的潜力。因此，从u的单次测量，我们得到，通过1/u，明确的界限上的指数衰减的系统的本征函数和估计的分布特征值附近的底部的频谱。
We consider the localization of eigenfunctions for the operator on a Lipschitz domain omega and, more generally, on manifolds with and without boundary. In earlier work, two authors of the present paper demonstrated the remarkable ability of the landscape, defined as the solution to Lu = 1, to predict the location of the localized eigenfunctions. Here, we explain and justify a new framework that reveals a richly detailed portrait of the eigenfunctions and eigenvalues. We show that the reciprocal of the landscape function, 1/u, acts as an effective potential. Hence from the single measurement of u, we obtain, via 1/u, explicit bounds on the exponential decay of the eigenfunctions of the system and estimates on the distribution of eigenvalues near the bottom of the spectrum.