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 ${\ell }^{\infty }$-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 ${\mathbb{Z}}^d$, 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.

我们证明了状态密度对无限图上离散、确定性薛定谔算子的势函数的依赖性的定量界限。虽然之前的结果仅限于具有独立同分布势的随机薛定谔算子，但本文开发了一个确定性框架，它适用于与势的特定性质无关的薛定谔算子。遵循 Bourgain 和 Klein 的想法，我们考虑了状态密度外测度 (DOSoM)，它对所有（确定性）薛定谔算子都有很好的定义。我们通过证明连续性的模数明确量化了 DOSoM 对电位的依赖性${\ell }^{\infty }$-规范。如此获得的特定连续性模量反映了无限远下图的几何形状。对于薛定谔算子的特殊情况${\mathbb{Z}}^d$，这意味着 DOSoM 的 Lipschitz 连续性关于电势。对于 Bethe 格子上的 Schrödinger 算子，我们获得了 DOSoM 对电位的 log-Hölder 依赖性。作为我们确定性框架的一个重要结果，我们获得了潜在采样函数中遍历薛定谔算子的状态密度度量 (DOSm) 的连续性模数。最后，我们通过使用与随机势相关的分位数函数在遍历框架中制定这个问题，恢复随机薛定谔算子对单点概率度量的依赖的随机薛定谔算子的先前结果。