Shiwen Zhang


U Minnesota


Friday, November 15, 2019 - 1:00pm to 2:00pm


rh 340p

In 2012, Filoche and Mayboroda introduced the concept of the landscape function u, for an elliptic operator L, which solves the inhomogeneous equation Lu=1. This landscape function has remarkable power to predict the shape and location of localized low energy eigenfunction. These ideas led to beautiful results in mathematics, as well as theoretical and experimental physics. In this talk, we first briefly review these results of landscape theory for differential operators on R^d. We will then discuss some recent progress of extending landscape theory to tight-binding Hamiltonians on discrete lattice Z^d. In particular, we show that the effective potential 1/u creates barrier for appropriate exponential decay eigenfunctions of Agmon type for some discrete Schrodinger operators. We also show that the minimum of 1/u leads to a new counting function, which gives non-asymptotic estimates on the integrated density of states of the Schrodinger operators. This talk contains joint work in progress with S. Mayboroda and some numerical experiments with W. Wang.