Department of Mathematics

Mathematical Physics and Operator Algebras

  •  Shiwen Zhang, U Minnesota
  •  The landscape law for tight binding Hamiltonians
  •  03/09/2021
  •  3:00 PM - 4:00 PM
  •   (Virtual Meeting Link)
  •  Jeffrey Hudson Schenker (

The localization landscape theory, introduced in 2012 by Filoche and Mayboroda, considers the so-called 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 also discuss some numerical experiments in progress on the so-called practical landscape law for the continuous Anderson model. This talk is based on joint work with D. N. Arnold, M. Filoche, S. Mayboroda, and Wei Wang.



Department of Mathematics
Michigan State University
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