Virtual Bound Levels in a Gap of the Essential Spectrum of the Weakly Perturbed Periodic Schrödinger Operator

In the space \({L_{2}(\mathbf{R}^{d}) (d \le 3)}\) we consider the Schrödinger operator \({H_{\gamma}=-{\Delta}+ V(\mathbf{x})\cdot+\gamma W(\mathbf{x})\cdot}\), where \({V(\mathbf{x})=V(x_{1}, x_{2}, \dots, x_{d})}\) is a periodic function with respect to all the variables, \({\gamma}\) is a small real coupling constant and the perturbation \({W(\mathbf{x})}\) tends to zero sufficiently fast as \({|\mathbf{x}|\rightarrow\infty}\). We study so called virtual bound levels of the operator \({H_\gamma}\), i.e., those eigenvalues of \({H_\gamma}\) which are born at the moment \({\gamma=0}\) in a gap \({(\lambda_-,\,\lambda_+)}\) of the spectrum of the unperturbed operator \({H_0=-\Delta+ V(\mathbf{x})\cdot}\) from an edge of this gap while \({\gamma}\) increases or decreases. We assume that the dispersion function of H₀, branching from an edge of \({(\lambda_-,\lambda_+)}\), is non-degenerate in the Morse sense at its extremal set. For a definite perturbation \({(W(\mathbf{x})\ge 0)}\) we show that if d ≤ 2, then in the gap there exist virtual eigenvalues which are born from this edge. We investigate their number and an asymptotic behavior of them and of the corresponding eigenfunctions as \({\gamma\rightarrow 0}\). For an indefinite perturbation we estimate the multiplicity of virtual bound levels. In particular, we show that if d = 3 and both edges of the gap \({(\lambda_-,\,\lambda_+)}\) are non-degenerate, then under additional conditions there is a threshold for the birth of the impurity spectrum in the gap, i.e., \({\sigma(H_\gamma)\cap(\lambda_-,\,\lambda_+)=\emptyset}\) for a small enough \({|\gamma|}\).     

Reversed‐phase ion‐pair high‐performance liquid chromatography assay of polyprenyl diphosphate oligomer homologues

A reversed‐phase ion‐pair high‐performance liquid chromatography procedure was developed for the separation of polyprenyl diphosphate oligomer homologues obtained chemically from plant polyprenols. Tetrabutylammonium phosphate was used as the ion‐pair reagent, and the dependence of the separation quality on pH of ion‐pair reagent was investigated for the first time. The procedure is applicable for the control of commercial available polyprenyl monophosphates (the active components of veterinary drugs Phosprenyl and Gamapren) for the possible presence of polyprenyl diphosphate byproducts.     

Cooperativity in ordinary ice and breaking of hydrogen bonds

The total interaction energy between two H-bonded water molecules in a condensed phase is composed of a binding energy between them and an energy due to a cooperative effect. An approximate simple expression is suggested for the dependence of the interaction energy between two H-bonded water molecules on the number of neighboring water molecules with which they are H-bonded. Using this expression, the probabilities of breaking a H bond with various numbers of H-bonded neighbors are estimated. These probabilities are used in computer simulations of the breaking of specified fractions of H bonds in an ordinary (hexagonal) ice. A large "piece" of hexagonal ice (up to 8 millions molecules) is built up, and various percentages of H bonds are considered broken. It is shown that 62-63% of H bonds must be broken in order to disintegrate the "piece" of ice into disconnected clusters. This value is only a little larger than the percolation threshold (61%) predicted both by the percolation theory for tetrahedral ice and by simulations in which all H bonds were considered equally probable to be broken. When the percentage of broken bonds is smaller than 62-63%, there is a network of H-bonded molecules which contains the overwhelming majority of water molecules. This result contradicts some models of water which consider that water consists of a mixture of water clusters of various sizes. The distribution of water molecules with unequal probabilities for breaking is compared with the simulation involving equal probabilities for breaking. It was found that in the former case, there is an enhanced number of water monomers without H bonds, that the numbers of 2- and 3-bonded molecules are smaller, and the number of 4-bonded molecules is larger than in the latter case.     

Stereochemical study of iminodihydrofurans based on experimental measurements and SOPPA calculations of (13)C-(13)C spin-spin coupling constants

Configurational assignment and conformational analysis of a series of iminodihydrofurans obtained from cyanoacetylenic alcohols were performed on the basis of experimental measurements and high-level ab initio calculations of their (13)C-(13)C spin-spin coupling constants. The title compounds were shown to form and exist in solution as the individual Z isomers, adopting the orthogonal orientation of the amino, alkylamino and dialkylamino groups and the s-trans orientation of the CONH(2) group at the C(4) position of the 2,5-dihydro-2-iminofuran moiety.     

Configurational assignment of N-arylsulfonylimines of alpha-polychloroaldehydes

Configurational assignment of seven synthesized N-arylsulfonylimines of alpha-polychloroaldehydes has been carried out by means of experimental measurements and high-level ab initio calculations of their (13)C--(13)C, (13)C--(1)H and (15)N--(1)H spin-spin coupling constants. The title compounds were shown to exist in solution solely in the form of E isomers, in line with thermodynamic reasoning.     

Hydrogen bonding between formic acid and water: complete stabilization of the intrinsically unstable conformer

We studied hydrogen bonding between formic acid (FA) and water in solid argon and identified the first water complex with the higher-energy conformer cis-FA. In sharp contrast to cis-FA monomer, cis-FA interacting with water is very stable at low temperatures, which was explained by strong O-H...O hydrogen bonding. These benchmark results show that hydrogen bonding can terminate proton tunneling reactions and efficiently stabilize intrinsically unstable conformational structures in complex asymmetrical hydrogen-bonded networks. This general effect occurs when the energy difference between conformers is smaller than the hydrogen bond interaction energy, which opens perspectives in chemistry on intrinsically unstable conformers.     

The first synthesis of furazano[3,4‐b][1,6]naphthyridines

Reaction of 6‐acetyl‐7‐aminofurazano[3,4‐b]pyridines with DMFDMA afforded N,N‐dimethylformamidines that were cyclized to the novel furazan‐fused [1,6]naphthyridine system by treatment with sodium methylate in good yield. The tricyclic system is characterized by X‐ray crystallography.     

Simple and fast method for the fabrication of switchable bicomponent micropatterned polymer surfaces

We report on the fabrication of micropatterned polymer surfaces that allow the reversible inversion of surface topography, charge, and wettability. Micropatterned surfaces were prepared by grafting two oppositely charged polyelectrolytes (poly(acrylic acid) and poly(2-vinylpyridine)) using a combination of photolithography, "lift off", and "grafting to" techniques. The switchable surfaces are of interest in microprinting and for the design of microfluidic devices and programmed protein adsorption.     

Use of Transmission Electron Microscopy in Combinatorial Studies of Functional Oxides

Summary: The principles of combinatorial methodology are based on high‐throughput properties measurements (HTPM) of multiple compositions in combinatorial libraries, and are recently being increasingly applied in materials research. Nevertheless, the authors' view is that the involvement of detailed but time‐consuming investigation using transmission electron microscopy (TEM) should be an important part of combinatorial materials research. In this paper we present three examples from our combinatorial studies where the TEM investigation was essential in obtaining a detailed picture of microstructures and their relationship with the physical properties. In the study of microwave dielectrics such as BaTiO₃‐SrTiO₃, TEM provides essential information on the type and distribution of defects in the deposited films. In the case of the wide‐band semiconductor ZnO‐MgO, the distribution and morphology of the phases were studied and related to the measured electronic properties. Study of the manganates LaMnO₃‐CaMnO₃ with colossal magneto‐resistive properties showed an anisotropic distribution of the structural domains and the morphology of the film. The distribution of the domains and the absence of epitaxial stresses found are essential in the interpretation of magnetic measurements.

Cross‐sectional TEM (dark field) micrograph taken with the reflection of h‐(Zn,Mg)O at x = 0.5.     

Hamiltonian loops from the ergodic point of view

Let G be the group of Hamiltonian diffeomorphisms of a closed symplectic manifold Y. A loop h:S¹→G is called strictly ergodic if for some irrational number α the associated skew product map T:S¹×Y→S¹×Y defined by T(t,y)=(t+α,h(t)y) is strictly ergodic. In the present paper we address the following question. Which elements of the fundamental group of G can be represented by strictly ergodic loops? We prove existence of contractible strictly ergodic loops for a wide class of symplectic manifolds (for instance for simply connected ones). Further, we find a restriction on the homotopy classes of smooth strictly ergodic loops in the framework of Hofer’s bi-invariant geometry on G. Namely, we prove that their asymptotic Hofer’s norm must vanish. This result provides a link between ergodic theory and symplectic topology. Received July 7, 1998 / final version received September 14, 1998