Dependence of the density of states outer measure on the potential for deterministic Schrödinger operators on graphs with applications to ergodic and random models
Peter D. Hislop, Christoph A. Marx
备注：
Related to arXiv:1804.02444 and arXiv:1904.01118 by the authors; New appendices C and D are added and typos corrected. Appendix C discusses inequalities between the metric of weak convergence of measures and the Kantorovich-Rubinstein-Wasserstein metric. Appendix D presents results on continuity of the Hausdorff distance between two spectra with respect to the potentials
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.
On the ground state of the nonlinear Schr{ö}dinger equation: asymptotic behavior at the endpoint powers
CNRS IRMAR - UMR 6625（法国国家科学研究中心 IRMAR）；Univ. Lille, CNRS, Inria UMR 8524 - Laboratoire Paul Painlevé（里尔大学，法国国家科学研究中心，Inria UMR 8524 - 保罗·庞加莱实验室）；Department of Mathematics, McMaster University（麦克马斯特大学数学系）
On unique continuation for solutions of the Schr{ö}dinger equation on trees
我们证明了如果齐次树上的含时薛定谔方程的解在两个不同的时刻快速衰减，则该解是平凡的.对于自由薛定谔算子，我们利用Laplacian谱理论和复分析，得到了导致在任意时刻急剧衰减的初始条件的一个特征.然后，我们利用Colin de Verdi和Turc的紧支势的Schr dinger算子的谱分解，推广了我们在紧支势存在下的结果.最后，我们使用实变量的方法首先介绍了Escauriaza，Kenig，Ponce和Vega建立一个一般的尖锐的结果的情况下，有界的潜力。
