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