Dependence of the density of states outer measure on the potential for deterministic Schrodinger operators on graphs with applications to ergodic and random models

We prove quantitative bounds on the dependence of the density of states on the potential function for discrete, deterministic Schrodinger operators on infinite graphs. While previous results were limited to random Schrodinger operators with independent, identically distributed potentials, this paper develops a deterministic framework, which is applicable to Schrodinger operators independent of the specific nature of the potential. Following ideas by Bourgain and Klein, we consider the density of states outer measure (DOSoM), which is well defined for all (deterministic) Schrodinger operators. We explicitly quantify the dependence of the DOSoM on the potential by proving a modulus of continuity in the l(infinity)-norm. The specific modulus of continuity so obtained reflects the geometry of the underlying graph at infinity. For the special case of Schrodinger operators on Z(d), this implies the Lipschitz continuity of the DOSoM with respect to the potential. For Schrodinger operators on the Bethe lattice, we obtain logHolder dependence of the DOSoM on the potential. As an important consequence of our deterministic framework, we obtain a modulus of continuity for the density of states measure (DOSm) of ergodic Schrodinger operators in the underlying potential sampling function. Finally, we recover previous results for random Schrodinger operators on the dependence of the DOSm on the single-site probability measure by formulating this problem in the ergodic framework using the quantile function associated with the random potential. (C) 2021 Elsevier Inc. All rights reserved.