TY - JOUR
T1 - Dependence of the density of states outer measure on the potential for deterministic Schrödinger operators on graphs with applications to ergodic and random models
AU - Hislop, Peter D.
AU - Marx, Christoph A.
N1 - Publisher Copyright:
© 2021 Elsevier Inc.
PY - 2021/11/1
Y1 - 2021/11/1
N2 - We prove quantitative bounds on the dependence of the density of states on the potential function for discrete, deterministic Schrödinger operators on infinite graphs. While previous results were limited to random Schrödinger operators with independent, identically distributed potentials, this paper develops a deterministic framework, which is applicable to Schrödinger 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) Schrödinger operators. We explicitly quantify the dependence of the DOSoM on the potential by proving a modulus of continuity in the ℓ∞-norm. The specific modulus of continuity so obtained reflects the geometry of the underlying graph at infinity. For the special case of Schrödinger operators on Zd, this implies the Lipschitz continuity of the DOSoM with respect to the potential. For Schrödinger operators on the Bethe lattice, we obtain log-Hölder 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 Schrödinger operators in the underlying potential sampling function. Finally, we recover previous results for random Schrödinger 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.
AB - We prove quantitative bounds on the dependence of the density of states on the potential function for discrete, deterministic Schrödinger operators on infinite graphs. While previous results were limited to random Schrödinger operators with independent, identically distributed potentials, this paper develops a deterministic framework, which is applicable to Schrödinger 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) Schrödinger operators. We explicitly quantify the dependence of the DOSoM on the potential by proving a modulus of continuity in the ℓ∞-norm. The specific modulus of continuity so obtained reflects the geometry of the underlying graph at infinity. For the special case of Schrödinger operators on Zd, this implies the Lipschitz continuity of the DOSoM with respect to the potential. For Schrödinger operators on the Bethe lattice, we obtain log-Hölder 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 Schrödinger operators in the underlying potential sampling function. Finally, we recover previous results for random Schrödinger 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.
KW - Density of states
KW - Ergodic Schrödinger operators
KW - Random Schrödinger operators
KW - Schrödinger operators
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U2 - 10.1016/j.jfa.2021.109186
DO - 10.1016/j.jfa.2021.109186
M3 - Article
AN - SCOPUS:85111234159
SN - 0022-1236
VL - 281
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 9
M1 - 109186
ER -