Controllability of the Schr\"odinger equation on unbounded domains without geometric control condition

We prove controllability of the Schr\"odinger equation in $\mathbb{R}^d$ in any time $T > 0$ with internal control supported on nonempty, periodic, open sets. This demonstrates in particular that controllability of the Schr\"odinger equation in full space holds for a strictly larger class of control supports than for the wave equation and suggests that the control theory of Schr\"odinger equation in full space might be closer to the diffusive nature of the heat equation than to the ballistic nature of the wave equation. Our results are based on a combination of Floquet-Bloch theory with Ingham-type estimates on lacunary Fourier series.Comment: 12 pages, 3 figures. We removed partly erroneous statements on fractional Laplacian

Sharp estimates and homogenization of the control cost of the heat equation on large domains

We prove new bounds on the control cost for the abstract heat equation, assuming a spectral inequality or uncertainty relation for spectral projectors. In particular, we specify quantitatively how upper bounds on the control cost depend on the constants in the spectral inequality. This is then applied to the heat flow on bounded and unbounded domains modeled by a Schr\"odinger semigroup. This means that the heat evolution generator is allowed to contain a potential term. The observability/control set is assumed to obey an equidistribution or a thickness condition, depending on the context. Complementary lower bounds and examples show that our control cost estimates are sharp in certain asymptotic regimes. One of these is dubbed homogenization regime and corresponds to the situation that the control set becomes more and more evenly distributed throughout the domain while its density remains constant.Comment: 28 pages, 3 figure

Magnetic Bernstein inequalities and spectral inequality on thick sets for the Landau operator

We prove a spectral inequality for the Landau operator. This means that for all $f$ in the spectral subspace corresponding to energies up to $E$, the $L^2$-integral over suitable $S \subset \mathbb{R}^2$ can be lower bounded by an explicit constant times the $L^2$-norm of $f$ itself. We identify the class of all measurable sets $S \subset \mathbb{R}^2$ for which such an inequality can hold, namely so-called thick or relatively dense sets, and deduce an asymptotically optimal expression for the constant in terms of the energy, the magnetic field strength and in terms of parameters determining the thick set $S$. Our proofs rely on so-called magnetic Bernstein inequalities. As a consequence, we obtain the first proof of null-controllability for the magnetic heat equation (with sharp bound on the control cost), and can relax assumptions in existing proofs of Anderson localization in the continuum alloy-type model.Comment: 26 page

Band Edge Localization Beyond Regular Floquet Eigenvalues

We prove that localization near band edges of multi-dimensional ergodic random Schr\"odinger operators with periodic background potential in $L^2(\mathbb{R}^d)$ is universal. By this we mean that localization in its strongest dynamical form holds without extra assumptions on the random variables and independently of regularity or degeneracy of the Floquet eigenvalues of the background operator. The main novelty is an initial scale estimate the proof of which avoids Floquet theory altogether and uses instead an interplay between quantitative unique continuation and large deviation estimates. Furthermore, our reasoning is sufficiently flexible to prove this initial scale estimate in a non-ergodic setting, which promises to be an ingredient for understanding band edge localization also in these situations.Comment: 13 pages; now the stronger dynamical localization in the short range case is formulated, a few references have beed added, minor editorial change

Photo-mediated oxygen atom transfer and light-induced organocatalytic and transition metal-catalyzed amination reactions

Die Arbeit beschreibt die Desoxygenierung verschiedener Pyrimidopteridin-N-Oxid-Derivate, deren photophysikalischen Untersuchungen sowie deren unterschiedliche Anwendung und mechanistischen Untersuchungen der metallfreie Hydroaminierung. Zusätzlich werden photophysikalische und elektrochemischen Eigenschaften von Pyridin-N-Oxid-Derivaten als Sauerstofftransferreagenz untersucht und die Umwandlung in das (Methylsulfonyloxy)pyridiniummethansulfonat beschrieben. Zuletzt wird eine Palladium-katalysierte C-N-Bindungsbildung zwischen Dimethylaminen und Aryltriflaten diskutiert

Veränderung der Effizienz der Regenwurmaustreibung mit Senfsuspensionen im Tagesverlauf

We tested the effect of sampling time on the efficiency of mustard extractions for earthworms in a field study. On an organic experimental farm (alluvial loams; Hennef/Germany) earthworm extractions were started on two consecutive days hourly from 10 am to 5 pm and 6 pm respectively. Significant effects of daytime on extracted earthworm biomass occurred on both days. Maximum biomass was extracted in the early afternoon in each case. Additional pot experiments have shown a significant temperature dependence of the mustard extraction method which at least partially explains the differences in on-site extracted earthworm biomass in the course of the day. We conclude that daytime has to be considered as a factor when using mustard extraction methods. When used in factorial field experiments, the extractions in different field plots should therefore be operated simultaneously