Localization of eigenfunctions via an effective potential

We consider the localization of eigenfunctions for the operator L = -div A grad + V on a Lipschitz domain Ω and, more generally, on manifolds with and without boundary. In earlier work, two authors of the present paper demonstrated the remarkable ability of the landscape, defined as the solution to Lu = 1, to predict the location of the localized eigenfunctions. Here, we explain and justify a new framework that reveals a richly detailed portrait of the eigenfunctions and eigenvalues. We show that the reciprocal of the landscape function, 1/u, acts as an effective potential. Hence from the single measurement of u, we obtain, via 1/u, explicit bounds on the exponential decay of the eigenfunctions of the system and estimates on the distribution of eigenvalues near the bottom of the spectrum.