论文标题
状态的密度外部度量对确定性Schrödinger操作员在图形上的潜力的依赖性，并应用了阵亡的模型和随机模型
Dependence of the density of states outer measure on the potential for deterministic Schrödinger operators on graphs with applications to ergodic and random models
论文作者
论文摘要
我们继续研究Schrödinger操作员对潜力的状态测量密度和相关光谱功能的依赖性。尽管我们较早的工作着重于随机Schrödinger运营商，但我们将这些结果扩展到具有确定性潜力和奇异潜力的无限图上的Schrödinger运营商，并改善了我们的随机电位结果。特别是，我们证明了在晶格上随机schrödinger操作员的DOSM的Lipschitz连续性，从而恢复了\ cite {kachkkovskiy，shamis}的结果。为了治疗确定性潜力，我们首先研究了为所有Schrödinger操作员定义的状态外措施密度（偶然性），并证明了剂量连续性相对于潜力的连续性模量的确定性结果。我们将这些结果应用于晶格$ Z^d $和Bethe Lattice的Schrödinger经营者。在前一种情况下，我们证明了阶段的Lipschitz连续性，在后一种情况下，我们证明了阶段为$ \ frac {1} {2} {2} $ - log-log-hölder连续。我们的技术结合了一个自我伴侣运算符的单参数家族的抽象Lipschitz特性与新的有限范围降低，使我们能够研究剂量和相关功能仅对有限量的变量的依赖性，并捕获了Infinity图表的几何形状。
We continue our study of the dependence of the density of states measure and related spectral functions of Schrödinger operators on the potential. Whereas our earlier work focused on random Schrödinger operators, we extend these results to Schrödinger operators on infinite graphs with deterministic potentials and ergodic potentials, and improve our results for random potentials. In particular, we prove the Lipschitz continuity of the DOSm for random Schrödinger operators on the lattice, recovering results of \cite{kachkovskiy, shamis}. For our treatment of deterministic potentials, we first study the density of states outer measure (DOSoM), defined for all Schrödinger operators, and prove a deterministic result of the modulus of continuity of the DOSoM with respect to the potential. We apply these results to Schrödinger operators on the lattice $Z^d$ and the Bethe lattice. In the former case, we prove the Lipschitz continuity of the DOSoM, and in the latter case, we prove that the DOSoM is $\frac{1}{2}$-log-Hölder continuous. Our technique combines the abstract Lipschitz property of one-parameter families of self-adjoint operators with a new finite-range reduction that allows us to study the dependency of the DOSoM and related functions on only finitely-many variables and captures the geometry of the graph at infinity.
