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.

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

## Location:

Ross 70