Shiwen Zhang (University of Minnesota)
Abstract: The localization landscape theory, introduced in 2012 by Filoche and Mayboroda, considers the landscape function u solving Hu=1 for an operator H. The landscape theory has remarkable power in studying the eigenvalue problems of H and has led to numerous ``landscape baked’’ results in mathematics, as well as in theoretical and experimental physics. In this talk, we will discuss some recent results of the landscape theory for tight-binding Hamiltonians H=-\Delta+V on Z^d. We introduce a box counting function, defined through the discrete landscape function of H. For any deterministic bounded potential, we give estimates for the integrated density of states from above and below by the landscape box counting function, which we call the landscape law. For the Anderson model, we get a refined lower bound for the IDS, throughout the spectrum. We will then discuss some numerical experiments on estimating eigenvalues via the landscape functions for the continuous Anderson model. In particular, we study the ground state energy of a Schrodinger operator and its relation to the landscape potential. For the 1-d Bernoulli Anderson model, we show that the ratio the ground state energy and the minimum of the landscape potential approaches to pi^2/8 as the domain size approaches infinity. We will also discuss some numerical stimulations and conjectures for excited states and for other random potentials. The talk is partially based on joint work with D. N. Arnold, I. Chenn, M. Filoche, S. Mayboroda, and Wei Wang.