# Dynamics and probability: David Jerison (MIT) - Localization of eigenfunctions via an effective potential

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.

## Date:

Tue, 18/06/2019 - 14:00 to 15:00

Ross 70