## Mathematical Physics and Harmonic Analysis Seminar

**Date: ** January 29, 2021

**Time: ** 2:00PM - 2:50PM

**Location: ** Zoom

**Speaker: **Chris Marx, Oberlin College

**Title: ***Potential dependence of the density of states: deterministic, ergodic, and random potentials*

**Abstract: **In this talk we will address the potential dependence of the density of states and related spectral functions for discrete Schr\"odinger operators on infinite graphs. Following ideas by J. Bourgain and A. Klein, we will consider the density of states {\em{outer}} measure (DOSoM), a {\em{deterministic}} quantity, which is well defined for {\em{all}} Schr\"odinger operators.
We will explicitly quantify the potential dependence of the DOSoM in weak topology by proving a modulus of continuity with respect to the potential in $\ell{l}^\infinity$-norm. The resulting modulus of continuity reflects the geometry of the graph at infinity. For the special case of operators on $\mathbb{Z}^d$ our result implies Lipschitz continuity of the DOSoM, in the case of the Bethe lattice, we obtain that the DOSoM is $\frac{1}{2}$-log-H\"older continuous. Applications of this result to ergodic, and in further consequence, for random Schr\"odinger operators will be presented.
This talk is based on joint work with Peter Hislop (University of Kentucky).