The hidden landscape of localization
Speaker: Svitlana Mayboroda, Univ. Minnesota
Location: Warren Weaver Hall 1302
Date: Thursday, April 12, 2018, 11 a.m.
Complexity of the geometry, randomness of the potential, and many other
irregularities of the system can cause powerful, albeit quite different,
manifestations of localization: a phenomenon of confinement of waves, or
eigenfunctions, to a small portion of the original domain. In the present
talk we show that behind a possibly disordered system there exists a clear
structure, referred to as a landscape function, which predicts the location
and shape of the localized eigenfunctions, a pattern of their exponential
decay, and delivers accurate bounds for the corresponding eigenvalues in the
range where, for instance, Weyl law notoriously fails. We will discuss main
features of this structure universally relevant for all elliptic operators,
as well as specific applications to the Schrodinger operator with random
potential and to the Poisson-Schrodinger drift-diffusion system.