Abstract We consider the localization of eigenfunctions for the operator on a Lipschitz domain Ω 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.

中文翻译：

---

通过有效势定位本征函数

摘要 我们考虑了在 Lipschitz 域 Ω 上的算子的特征函数的定位，更一般地说，在有边界和无边界的流形上。在早期的工作中，本文的两位作者展示了景观的非凡能力，定义为 Lu = 1 的解，预测局部特征函数的位置。在这里，我们解释并证明了一个新框架的合理性，该框架揭示了特征函数和特征值的丰富详细的描述。我们表明景观函数的倒数 1/u 作为有效势。因此，从 u 的单次测量中，我们通过 1/u 获得系统特征函数指数衰减的明确界限，并估计频谱底部附近的特征值分布。