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

TL;DR

This paper reviews and extends the use of landscape functions for eigenvector bounds of Schrödinger operators, applying to various operators and deriving eigenvalue and heat kernel estimates with practical examples.

Contribution

It unifies existing approaches to landscape function bounds, extends them to new operators, and derives novel eigenvalue and heat kernel estimates using order properties.

Findings

Unified several approaches for eigenvector bounds

Extended landscape function methods to nonlinear operators

Derived lower bounds on principal eigenvalues and upper bounds on heat kernels

Abstract

Landscape functions are a popular tool used to provide upper bounds for eigenvectors of Schr\"odinger 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. We also use landscape functions to derive lower estimates on the principal eigenvalue -- much in the spirit of earlier results by Donsker-Varadhan and Ba\~nuelos-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 principles. Additionally, we illustrate our findings with several examples, including p-Laplacians on domains and graphs as well as Schr\"odinger operators with magnetic and electric potential, also by means of elementary numerical experiments.