We discuss joint work with Douglas Arnold, Guy David, Marcel Filoche and Svitlana Mayboroda.
Consider for the operator $L = -\Delta + V$ with periodic boundary conditions, and more
generally on the manifold with or without boundary. Anderson localization, a significant feature
of semiconductor physics, says that the eigenfunctions of $L$ are exponentially localized with
high probability for many classes of random potentials $V$. Filoche and Mayboroda introduced the
function $u$ solving $Lu = 1$ and showed numerically that it strongly reflects this localization.
Here, we deepen the connection between the eigenfunctions and their landscape function $u$
by proving that its reciprocal $1/u$ acts as an effective potential. The effective potential
governs the exponential decay of the eigenfunctions and delivers information on the distribution
of eigenvalues near the bottom of the spectrum.