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

Abstract

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 ten years, unify several approaches used to achieve such bounds, and extend their scope to a large class of linear and nonlinear operators on general Banach lattices. We also use landscape functions to derive lower estimates on spectral gaps and Cheeger constants, 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.We illustrate our findings with several examples, including p-Laplacians on domains and graphs; the second derivative with Dirichlet boundary conditions; and Schrödinger operators with magnetic and electric potential.