Positivity and Negativity of Solutions to a Schrödinger Equation in ℝ N

Weak L ² -solutions u of the Schrödinger equation, –u + q(x) u – u = f(x) in L ² , are represented by a Fourier series using spherical harmonics in order to prove the following strong maximum and anti-maximum principles in (N 2): Let ₁ denote the positive eigenfunction associated with the principal eigenvalue ₁ of the Schrödinger operator . Assume that the potential q(x) is radially symmetric and grows fast enough near infinity, and f is a `sufficiently smooth' perturbation of a radially symmetric function, f 0 and 0 f / C const a.e. in . Then u is ₁-positive for - < < ₁ (i.e., u c ₁ with c const > 0) and ₁-negative for ₁ < < ₁ + (i.e., u –c₁ with c const > 0), where > 0 is a number depending on f. The constant c > 0 depends on both and f.     

Parametric generation of twin photons in vertical triple microcavities

We report the realization of a monolithic vertical-cavity, surface emitting micro-optical parametric conversion nanostructure, triply resonant with the parametric frequencies, allowing parametric oscillation with ultra-low pump power threshold. The photonic phase-space naturally provides triple resonance for the parametric frequencies, together with built-in cavity phase-matching for the pump wave at normal incidence. Parametric oscillation is observed in both the strong and weak exciton–photon coupling regime, allowing a high operating temperature. Signal and idler beams can be collected at 0° or at finite angles. The OPO threshold is low enough to envisage the realization of an all-semiconductor electrically-pumped micro-parametric oscillator. To cite this article: C. Diederichs et al., C. R. Physique 8 (2007).     

Static SIMS study on surfaces of chalcogenide glasses modified by an organic layer

Chalcogenide glasses are useful optic materials that find applications in infrared spectroscopy, sensors and thermal imaging. A route for direct surface modification of such glasses with organic layers has been investigated by static Time of Flight Secondary Ion Mass Spectrometry (ToF-SIMS). The GAS (germanium–arsenic–selenium) glasses are modified by deposition followed by UV-irradiation of disulfide- or/and silane-functionalized organic molecules. SIMS analysis shows the characteristic fragments of the grafted molecules and organic–inorganic fragments which prove unambiguously the binding mode to the surface: disulfides, after S–S cleavage, are linked to arsenic and selenium; triethoxysilanes bind exclusively to oxidized germanium. The successive grafting of disulfide and silane compounds on the same substrate (IG2 glass with 33% of Ge) affords a “mixed” organic layer on the glass surface. From water contact angle measurements and X-ray Photo-electron Spectroscopy (XPS), the coverage density is not significantly improved comparatively to the “non-mixed” layers. However, the grafting of both types of molecules allows to reach a more homogeneous coverage.     

Polypropylene ionic thermoplastic elastomers: Synthesis and properties

Polypropylene ionic thermoplastic elastomers have been prepared by melt radical grafting of maleic anhydride onto polypropylene in the presence of N-bromosuccinimide followed by neutralization of the resulting elastomeric grafted polypropylene using sodium salts. Sodium hydroxide and sodium acetate were compared in aqueous solution, as anhydrous or hydrated powders. The neutralization reaction was followed by Fourier transform infrared spectroscopy, allowing the development of a method to determine the effective neutralization degree. Important physical changes were recorded upon neutralization. Especially thermal stability, shear storage modulus and complex viscosity in the flow region were largely increased as a function of the neutralization degree.     

N‐vinylcaprolactam‐based microgels: Synthesis and characterization

Poly(N‐vinylcaprolactam) (PVCL) is well known for its thermoresponsive behavior in aqueous solutions. PVCL combines useful and important properties; it is biocompatible polymer with the phase transition in the region of physiological temperature (32–38 °C). This combination of properties allows consideration of PVCL as a material for design biomedical devices and use in drug delivery systems. In this work, PVCL based temperature‐sensitive crosslinked particles (microgels) were synthesized in a batch reactor to analyze the effect of the crosslinker, initiator, surfactant, temperature, and VCL concentration on polymerization process and final microgels characteristics. The mean particle diameters at different temperatures and the volume phase‐transition temperature of the final product were analyzed. To obtain information about the inner structure of microgel particles, semicontinuous polymerizations were carried out and the evolution of the hydrodynamic average particle diameters at different temperatures of the microgel synthesized was investigated. In the case of microgel particles obtained in a batch reactor the size and the swelling‐de‐swelling behavior as a function of the temperature of the medium can be tuned by modulating the reaction variables. When the microgel particles were synthesized in a semibatch reactor different swelling‐de‐swelling behaviors were observed depending on particles inner structure. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 2510–2524, 2008     

A Closer Look at 3,5-Dinitrotoluene

The dinitrotoluenes (DNT) are some widely used compounds despite the first general impression one might have. They are the intermediates of several economically important industrial reactions such as the continuous nitration process of trinitrotoluene (TNT).^3,4 As we were interested in the spectral characteristics of the various isomers of DNT⁵, we realized that the 3,5 isomer was not u-sually obtained during the nitration process of toluene⁶, thus making its availability quite restricted. It appeared relevant to find a useful and practical way to synthesize that particular isomer as it could also lead to the formation of new polyurethanes⁷, many polymerization processes uses DNT (mainly the 2,4 isomer because of its availability and its low price).     

Emulsion polymerization of styrene using polystyrene‐block‐poly(ethylene oxide) macromonomers as stabilizers

New diblock macromonomers were used as reactive emulsifiers in the emulsion polymerization of styrene. The nature of the reactive group, the molecular weight, the length of the poly(ethylene oxide) (PEO) block, and the molecular structure of the macromonomer were systematically investigated during this process by analyzing the evolution of the conversion and particle diameters. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 40: 2819–2827, 2002