Landscape functions are a popular tool used to provide upper bounds for eigenvectors of Schrödinger operators on domains. We review some known results obtained in the last 10 years, unify several approaches used to achieve such bounds, and extend their scope to a large class of linear and nonlinear operators. We also use landscape functions to derive lower estimates on the principal eigenvalue—much in the spirit of earlier results by Donsker–Varadhan and Bañuelos–Carrol—as well as upper bounds on heat kernels. Our methods solely rely on order properties of operators: We devote special attention to the case where the relevant operators enjoy various forms of elliptic or parabolic maximum principle. Additionally, we illustrate our findings with several examples, including p-Laplacians on domains and graphs as well as Schrödinger operators with magnetic and electric potential, also by means of elementary numerical experiments.

中文翻译：

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通过景观函数进行逐点特征向量估计：Filoche-Mayboroda-van den Berg 界的一些变化

景观函数是一种流行的工具，用于提供域上薛定谔算子的特征向量的上限。我们回顾了过去 10 年中获得的一些已知结果，统一了用于实现此类界限的几种方法，并将其范围扩展到一大类线性和非线性算子。我们还使用景观函数来得出主特征值的较低估计值（很大程度上符合 Donsker-Varadhan 和 Bañuelos-Carrol 早期结果的精神）以及热核的上限。我们的方法仅依赖于算子的序属性：我们特别关注相关算子享有各种形式的椭圆或抛物线极大值原理的情况。此外，我们还通过几个例子来说明我们的发现，包括域和图上的p拉普拉斯算子以及具有磁势和电势的薛定谔算子，也是通过基本数值实验来实现的。