Cwikel's bound reloaded

There are a couple of proofs by now for the famous Cwikel--Lieb--Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schr\"odinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel's proof is said to give a constant which is at least about 2 orders of magnitude off the truth. This situation did not change much during the last 40+ years. It turns out that this common belief, i.e, Cwikel's approach yields bad constants, is not set in stone: We give a drastic simplification of Cwikel's original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our proof is also quite flexible and leads to rather precise bounds for a large class of Schr\"odinger-type operators with generalized kinetic energies. Moreover, it highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis.Comment: 30 page

N,N'-dimethylperylene-3,4,9,10-bis(dicarboximide) on alkali halide(001) surfaces

The growth of N,N'-dimethylperylene-3,4,9,10-bis(dicarboximide) (DiMe-PTCDI) on KBr(001) and NaCl(001) surfaces has been studied. Experimental results have been achieved using frequency modulation atomic force microscopy at room temperature under ultra-high vacuum conditions. On both substrates, DiMe-PTCDI forms molecular wires with a width of 10 nm, typically, and a length of up to 600 nm at low coverages. All wires grow along the [110] direction (or [1$\bar{1}$0] direction, respectively) of the alkali halide (001) substrates. There is no wetting layer of molecules: Atomic resolution of the substrates can be achieved between the wires. The wires are mobile on KBr surface but substantially more stable on NaCl. A p(2 x 2) superstructure in brickwall arrangement on the ionic crystal surfaces is proposed based on electrostatic considerations. Calculations and Monte-Carlo simulations using empirical potentials reveal possible growth mechanisms for molecules within the first layer for both substrates, also showing a significantly higher binding energy for NaCl(001). For KBr, the p(2 x 2) superstructure is confirmed by the simulations, for NaCl, a less dense, incommensurate superstructure is predicted.Comment: 5 pages, 5 figure

Untersuchung der Adsorption von organischen Molekülen

Die vorliegende Arbeit untersucht die nicht kovalente Wechselwirkung von organischen Molekülen auf nicht organischen Oberflächen. Im Verlauf der Arbeit wurden zunächst die Voraussetzungen für die Präparation molekulare Schichten mit diesen Molekülen unter Ultrahochvakuumbedingungen untersucht. Es zeigte sich, dass beide Moleküle thermisch verdampft werden können, ohne, dass eine nennenswerte Zersetzung geschieht. Mit Hilfe thermischer Desorptionsspektroskopie wurde dann die Bindungsenergie der Moleküle auf der Nickeloberfläche bestimmt. Weiterhin konnte so auch die Bedeckung als Funktion der Aufdampfparameter ermittelt werden. Die räumliche Anordnung auf der Oberfläche sollte dann mit der Rastertunnelmikroskopie analysiert werden. Die chemische Umgebung der adsorbierten Moleküle wurde mittels Photoelektronenspektroskopie untersucht. In einem letzen Schritt wurde eine Heteroschicht von TCNQ auf Tweezer auf Nickel mittels thermischer Desorptionsspektroskopie untersucht. Hierbei stand die Frage im Vordergrund, ob sich der Komplex auch ohne Lösungsmittel auf einer Oberfläche bilden kann

Cwikel’s bound reloaded

There are a couple of proofs by now for the famous Cwikel–Lieb–Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schrödinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel’s proof is said to give a constant which is at least about 2 orders of magnitude off the truth. This situation did not change much during the last 40+ years. It turns out that this common belief, i.e, Cwikel’s approach yields bad constants, is not set in stone: We give a drastic simplification of Cwikel’s original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our proof is also quite flexible and leads to rather precise bounds for a large class of Schrödingertype operators with generalized kinetic energies. Moreover, it highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis

Cwikel’s bound reloaded

There are several proofs by now for the famous Cwikel–Lieb–Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schrödinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel’s proof is said to give a constant which is at least about 2 orders of magnitude off the truth. This situation did not change much during the last 40+ years. It turns out that this common belief, i.e, Cwikel’s approach yields bad constants, is not set in stone: We give a substantial refinement of Cwikel’s original approach which highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis. Moreover, it gives an astonishingly good bound for the constant in the CLR inequality. Our proof is also quite flexible and leads to rather precise bounds for a large class of Schrödinger-type operators with generalized kinetic energies

Temperature dependence of the energy dissipation in dynamic force microscopy

The dissipation of energy in dynamic force microscopy is usually described in terms of an adhesion hysteresis mechanism. This mechanism should become less efficient with increasing temperature. To verify this prediction we have measured topography and dissipation data with dynamic force microscopy in the temperature range from 100 K up to 300 K. We used 3,4,9,10-perylenetetracarboxylic-dianhydride (PTCDA) grown on KBr(001), both materials exhibiting a strong dissipation signal at large frequency shifts. At room temperature, the energy dissipated into the sample (or tip) is 1.9 eV/cycle for PTCDA and 2.7 eV/cycle for KBr, respectively, and is in good agreement with an adhesion hysteresis mechanism. The energy dissipation over the PTCDA surface decreases with increasing temperature yielding a negative temperature coefficient. For the KBr substrate, we find the opposite behaviour: an increase of dissipated energy with increasing temperature. While the negative temperature coefficient in case of PTCDA agrees rather well with the adhesion hysteresis model, the positive slope found for KBr points to a hitherto unknown dissipation mechanism

A dense network of cosmic-ray neutron sensors for soil moisture observation in a highly instrumented pre-Alpine headwater catchment in Germany

Monitoring soil moisture is still a challenge: it varies strongly in space and time and at various scales while conventional sensors typically suffer from small spatial support. With a sensor footprint up to several hectares, cosmic-ray neutron sensing (CRNS) is a modern technology to address that challenge. So far, the CRNS method has typically been applied with single sensors or in sparse national-scale networks. This study presents, for the first time, a dense network of 24 CRNS stations that covered, from May to July 2019, an area of just 1 km2: the pre-Alpine Rott headwater catchment in Southern Germany, which is characterized by strong soil moisture gradients in a heterogeneous landscape with forests and grasslands. With substantially overlapping sensor footprints, this network was designed to study root-zone soil moisture dynamics at the catchment scale. The observations of the dense CRNS network were complemented by extensive measurements that allow users to study soil moisture variability at various spatial scales: roving (mobile) CRNS units, remotely sensed thermal images from unmanned areal systems (UASs), permanent and temporary wireless sensor networks, profile probes, and comprehensive manual soil sampling. Since neutron counts are also affected by hydrogen pools other than soil moisture, vegetation biomass was monitored in forest and grassland patches, as well as meteorological variables; discharge and groundwater tables were recorded to support hydrological modeling experiments. As a result, we provide a unique and comprehensive data set to several research communities: to those who investigate the retrieval of soil moisture from cosmic-ray neutron sensing, to those who study the variability of soil moisture at different spatiotemporal scales, and to those who intend to better understand the role of root-zone soil moisture dynamics in the context of catchment and groundwater hydrology, as well as land–atmosphere exchange processes. The data set is available through the EUDAT Collaborative Data Infrastructure and is split into two subsets: https://doi.org/10.23728/b2share.282675586fb94f44ab2fd09da0856883 (Fersch et al., 2020a) and https://doi.org/10.23728/b2share.bd89f066c26a4507ad654e994153358b (Fersch et al., 2020b)

COSMOS-Europe : a European network of cosmic-ray neutron soil moisture sensors

We thank TERENO (Terrestrial Environmental Observatories), funded by the Helmholtz-Gemeinschaft for the financing and maintenance of CRNS stations. We acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) of the research unit FOR 2694 Cosmic Sense (grant no. 357874777) and by the German Federal Ministry of Education of the Research BioökonomieREVIER, Digitales Geosystem – Rheinisches Revier project (grant no. 031B0918A). COSMOS-UK has been supported financially by the UK’s Natural Environment Research Council (grant no. NE/R016429/1). The Olocau experimental watershed is partially supported by the Spanish Ministry of Science and Innovation through the research project TETISCHANGE (grant no. RTI2018-093717-BI00). The Calderona experimental site is partially supported by the Spanish Ministry of Science and Innovation through the research projects CEHYRFO-MED (grant no. CGL2017-86839- C3-2-R) and SILVADAPT.NET (grant no. RED2018-102719-T) and the LIFE project RESILIENT FORESTS (grant no. LIFE17 CCA/ES/000063). The University of Bristol’s Sheepdrove sites have been supported by the UK’s Natural Environment Research Council through a number of projects (grant nos. NE/M003086/1, NE/R004897/1, and NE/T005645/1) and by the International Atomic Energy Agency of the United Nations (grant no. CRP D12014). Acknowledgements. We thank Peter Strauss and Gerhab Rab from the Institute for Land and Water Management Research, Federal Agency for Water Management Austria, Petzenkirchen, Austria. We thank Trenton Franz from the School of Natural Resources, University of Nebraska–Lincoln, Lincoln, NE, United States. We also thank Carmen Zengerle, Mandy Kasner, Felix Pohl, and Solveig Landmark, UFZ Leipzig, for supporting field calibration, lab analysis, and data processing. We furthermore thank Daniel Dolfus, Marius Schmidt, Ansgar Weuthen, and Bernd Schilling, Forschungszentrum Jülich, Germany. The COSMOS-UK project team is thanked for making its data available to COSMOS-Europe. Luca Stevanato is thanked for the technical details about the Finapp sensor. The stations at Cunnersdorf, Lindenberg, and Harzgerode have been supported by Falk Böttcher, Frank Beyrich, and Petra Fude, German Weather Service (DWD). The Zerbst site has been supported by Getec Green Energy GmbH and Jörg Kachelmann (Meteologix AG). The CESBIO sites have been supported by the CNES TOSCA program. The ERA5-Land data are provided by ECMWF (Muñoz Sabater, 2021). The Jena dataset was retrieved at the site of The Jena Experiment, operated by DFG research unit FOR 1451.Peer reviewedPublisher PD