Wei Wang,University of Minnesota

Waves of all sorts, including electromagnetic waves, acoustic waves, and mechanical vibrations, play a role in shaping our world. In disordered systems, the transport of waves is hindered, resulting in a suppression of their motion. Building on the recent theoretical progress, the 'localization landscape' enables the solution of some of the most compelling puzzles of wave localizations in disordered media. In this talk, we will discuss random Schr?dinger operators. Using the effective potential, we show that the ratio between the eigenvalues and the local minima of the effective potential approaches constants as the domain size approaches infinity. By counting the local minima points of the effective potential, we introduce the landscape law to establish non-asymptotic estimates from above and below on the integrated density of states (IDOS). Further, we will discuss various improved approximations to the IDOS between energy ranges.