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Landscape functions are a popular tool used to provide upper bounds foreigenvectors of Schr\"odinger operators on domains. We review some knownresults obtained in the last ten years, unify several approaches used toachieve such bounds, and extend their scope to a large class of linear andnonlinear operators. We also use landscape functions to derive lower estimateson the principal eigenvalue -- much in the spirit of earlier results byDonsker-Varadhan and Ba\~nuelos-Carrol -- as well as upper bounds on heatkernels. Our methods solely rely on order properties of operators: we devotespecial attention to the case where the relevant operators enjoy various formsof elliptic or parabolic maximum principles. Additionally, we illustrate ourfindings with several examples, including p-Laplacians on domains and graphs aswell as Schr\"odinger operators with magnetic and electric potential, also bymeans of elementary numerical experiments.

Pointwise eigenvector estimates by landscape functions: some variations on the Filoche--Mayboroda--van den Berg bound