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

We continue our study of the dependence of the density of states measure and related spectral functions of Schr\"odinger operators on the potential. Whereas our earlier work focused on random Schr\"odinger operators, we extend these results to Schr\"odinger 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 Schr\"odinger 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 Schr\"odinger operators, and prove a deterministic result of the modulus of continuity of the DOSoM with respect to the potential. We apply these results to Schr\"odinger 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-H\"older 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.Comment: 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 potential

Manifestations of Local Supersolidity of 4^{4}He around a Charged Molecular Impurity

A frozen, solid helium core, dubbed snowball, is typically observed around cations in liquid helium. Here we discover, using path integral simulations, that around a cationic molecular impurity, protonated methane, the $^4$He atoms are indeed strongly localized akin to snowballs but still participate in vivid bosonic exchange induced by the ro-vibrational motion of the impurity. Such combination of solid-like order with pronounced superfluid response in the first helium shell indicates that manifestations of local supersolid behavior of $^4$He can be induced -- and probed experimentally -- by charged molecules

Trennung und Angst: Hendrik Verwoerd und die Gedankenwelt der Apartheid

Die Studie untersucht am Beispiel des Premierministers Hendrik Verwoerd (1958-66) die politische Gedankenwelt der Apartheid in Südafrika. Dabei wird erstmals der intellektuelle Werdegang Verwoerds mit seiner Politik der radikalen Rassentrennung in einer Kontinuität gesehen. Durch die politische Repression nach innen führte Verwoerd sein Land in die außenpolitische Isolation

Highly arc transitive digraphs

Unendliche, hochgradig bogentransitive Digraphen werden definiert und anhand von Beispielen vorgestellt. Die Erreichbarkeitsrelation und Eigenschaft–Z werden definiert und unter Verwendung von Knotengraden, Wachstum und anderen Eigenschaften, die von der Untersuchung von Nachkommen von Doppelstrahlen oder Automorphismengruppen herrühren, auf hochgradig bogentransitiven Digraphen untersucht. Seifters Theoreme über hochgradig bogentransitive Digraphen mit mehr als einem Ende, seine daherrührende Vermutung und deren sie widerlegende Gegenbeispiele werden vorgestellt. Eine Bedingung, unter der C–homogene Digraphen hochgradig bogentransitiv sind, wird angegeben und die Verbindung zwischen hochgradig bogentransitiven Digraphen und total unzusammenhängenden, topologischen Gruppen wird erwähnt. Einige Bemerkungen über die Vermutung von Cameron–Praeger–Wormald werden gemacht und eine verfeinerte Version vermutet. Die Eigenschaften der bekannten hochgradig bogentransitiven Digraphen werden gesammelt. Es wird festgestellt, dass einige, aber nicht alle unter ihnen Cayley–Graphen sind. Schließlich werden offen gebliebene Fragestellungen und Vermutungen zusammengefasst und neue hinzugefügt. Für die vorgestellten Lemmata, Propositionen und Theoreme sind entweder Beweise enthalten, oder Referenzen zu Beweisen werden angegeben.Infinite, highly arc transitive digraphs are defined and examples are given. The Reachability–Relation and Property-Z are defined and investigated on infinite, highly arc transitive digraphs using the valencies, spread and other properties arising from the investigation of the descendants of lines or the automorphism groups. Seifters theorems about highly arc transitive digraphs with more than one end, his conjecture on them and the counterexamples that disproved his conjecture, are given. A condition for C–homogeneous digraphs to be highly arc transitve is stated and the connection between highly arc transitive digraphs and totally disconnected, topological groups is mentioned. Some notes on the Cameron–Praeger–Wormald–Conjecture are made and a refined conjecture is stated. The properties of the known highly arc transitive digraphs are collected, some but not all of them are Cayley–graphs. Finally open questions and conjectures are stated and new ones are added. For the given lemmas, propositions and theorems either proofs or references to proofs are included