Abstract
We study localization properties of low-lying eigenfunctions for rapidly varying potentials V in bounded domains
Filoche & Mayboroda introduced the landscape function
and showed that the function u has remarkable properties: localized eigenfunctions prefer to localize in the local maxima of u. Arnold, David, Filoche, Jerison & Mayboroda showed that
arises naturally as the potential in a related equation. Motivated by these questions, we introduce a one-parameter family of regularized potentials Vt that arise from convolving V with the radial kernel
We prove that for eigenfunctions
this regularization Vt is, in a precise sense, the canonical effective potential on small scales. The landscape function u respects the same type of regularization. This allows allows us to derive landscape-type functions out of solutions of the equation
for a general right-hand side
2010 MATHEMATICS SUBJECT CLASSIFICATION: