On the Dirac–Kepler problem: The Johnson–Lippmann operator,supersymmetry, and normal-mode representations

The relativistic Kepler problem is discussed, with emphasis on the exact supersymmetry of the problem. It is shown that the supersymmetry is generated by the Johnson–Lippmann operator. Two related operators are found to generate new supersymmetries in an extended function space. Each of these supersymmetries may be disguised as radial supersymmetries. The radial supersymmetries are discussed and it is shown that each of them defines a normal-mode representation of the hydrogen-atom radial functions. Thus, one obtains two different, but equivalent, analytical expressions for these functions. The expressions are well known, but are rederived here in the light of the new understanding. Finally, the nonrelativistic image of the relativistic supersymmetry is constructed and its generators shown to be identical with those recently presented in the literature. © 1995 John Wiley & Sons, Inc.     

Tunneling of an energy eigenstate through a parabolic barrier viewed from Wigner phase space

We analyze the tunneling of a particle through a repulsive potential resulting from an inverted harmonic oscillator in the quantum mechanical phase space described by the Wigner function. In particular, we solve the partial differential equations in phase space determining the Wigner function of an energy eigenstate of the inverted oscillator. The reflection or transmission coefficients R or T are then given by the total weight of all classical phase-space trajectories corresponding to energies below, or above the top of the barrier given by the Wigner function.     

Synthesis and characterization of Di-π-cyclopentadienyl- metal pentasulfides of titanium(IV) and vanadium (IV):an operational test of the influence of an unpaired electron on the molecular geometry

Our preparation of Ti(h⁵?C₅H₅)₂ S₅ by the reaction of elemental sulfur with Ti(h⁵?C₅H₅)₂(CO)₂ in hexane and of V(h⁵?C₅H₅)₂S₅?$\text{1}\text{2}$ H₂O by the reacti of V(h⁵?C₅H₅)₂ Cl₂ with Na₂S₅ in THF and structural analyses by single crystal X-ray diffraction (together with infrared, solution EPR, and temperaturedependent magnetic susceptibility measurements) represent an extension of our previous work on M(h⁵?C₅H₅)₂ (SC₆H₅)₂ (M = Ti, V). The crystallographic results provide further support of our previous conclusions that the Ballhausen—Dahl model is not valid for M(h⁵?C₅H₅)₂L₂ systems. The structuralfeatures of the chair-like titanium and vanadium pentasulfide molecules are compared to the corresponding phenylmercapto analogs and to the chair-like cyclohexasulfur molecule in rhombohedral sulfur. Ti(h⁵?C₅H₅)₂S₅ was isolated as a mixture of monoclinic and orthorhombic Crystalline phases-which were both characterized by preliminary X-ray data. A complete Structural determination and refinement of the monoclinic phase, which contains two independent molecules in a cell of dimensions a 22.843(2), b 7.958(1), c 14.465(1) Å, β 90.074(4)° and symmetry P2₁/c, yielded R₁ 5.3 % and R₂ 5.9 % for 2168 independent diffractometry-collected data with I≥ 2.5o(I). V(h⁵?C₅H₅)₂S₅-$\text{1}\text{2}$ H₂O contains four V(h⁵?C₅H₅)₂S₅ molecules and two water molecules of hydration (of crystallographic site symmetry C₂-2) in onorthorhombic unit cell of symmetry P2₁2₁2 and of dimensions a 13.491(1), b 12.748(1), c 7.715(1) Å. Least-squares refinement of 750 diffractometry data with I≥2.0σ(I) gave R₁ 2.4% and R₂ 3.0% Both of these compounds were independently synthesized and Spectroscopically characterized by Köpf and co-workers, and-a Complete X-ray diffraction study was performed by Epstein and Bernal on a different monoclinic phase of Ti(h⁵?C₅H₅)₂S₅ (isolated by-Köpf). An extraction of V(h⁵?C₅H₅)₂S₅ with re fluxing benzene under nitrogen atmosphere in a Soxhlet apparatus led to the formation of the previously reported [V₂ (h⁵?C₅H₅)₂S₅]_n compound which was characterized by physical measurements including a preliminary X-ray diffraction study.     

The potentiometric determination of insecticides: Part I. Organophosphorus insecticides inhibiting cholinesterase

A potentiometric method for the determination of organophosphorus insecticides based on the inhibition of cholinesterase is presented. The acetic acid formed by hydrolysis of acetylcholine is sensed by a glass electrode in a weakly buffered system. The insecticide is incubated with cholinesterase for 1 h at 25°C or 37°C before addition to the substrate. The effects of incubation time and temperature are discussed. The method is applied to the insecticides bromophos and dichlorvos. The detection limits are 2 × 10^-5 M (7 ppb) for bromophos and 1 × 10^-7 M (22 ppb) for dichlorvos. The initial rates of hydrolysis decrease linearly up to 2 × 10^-7 M and 1 × 10^-6 M, respectively.     

DERIVATIVES OF 4-CHLORO-3,5-DINITROBENZOTRIFLUORIDE V. SYNTHESIS OF 4,6-DINITRO-2,8-BIS(TRIFLUOROMETHYL) PHENOTHIAZINE (1), ISOPROPYL 2,6-DINITRO-4-TRIFLUOROMETHYLPHENYL SULFIDE (2) AND BIS(2,6-DINITRO-4-TRIFLUOROMETHYLPHENYL) DISULFIDE (3)

Abstract

The reaction of potassium isopropyl dithiocarbonate with 4-chloro-3,5-dinitrobenzotrifluoride in dimethylformamide at 25–30°C afforded the titled phenothiazine (1) and sulfide (2). The reaction of bis(2,6-dinitro-4-trifluoromethylphenyl)-sulfide or the above halogen compound with sodium hydrosulfide in the same solvent at 80–90°C furnished 1. At 25–30°C the latter reaction afforded the titled disulfide (3). Possible mechanisms and supporting nmr, ir and mass spectral data are discussed.     

Graphs with two isomorphism classes of spanning unicyclic subgraphs

The graphs whose spanning unicyclic subgraphs partition into exactly two isomorphism classes are characterized.This work is a continuation of [6] where graphs with one isomorphism class of spanning unicyclic graphs are characterized. The analogous question for spanning trees was posed in [10] and graphs with one isomorphism class of spanning trees were characterized in [2], [3], [4], [7], [11] while graphs with two isomorphism classes of spanning trees were characterized in [4], [5]. Related topics are treated in [1], [8], [9].     

Homologies cyclique et de Hochschild de certains espaces homogènes quantiques

Nous quantifions certaines inclusions d'algèbres de Lie semi-simpleshg. Nous calculons les homologies associées aux quantifications, surC((h)), d'une part des algèbres de fonctions formelles surG/H, pourHG une inclusion de groupes de Lie semi-simples associée, et d'autre part des fonctions algébriques sur SL(2,C)/T.We quantize certain inclusions of semisimple Lie algebrashg. We compute the cyclic and Hochschild homologies for theC((h))-quantizations of

Prescribing eigenvalues of the Dirac operator

In this note we show that every compact spin manifold of dimension ≥3 can be given a Riemannian metric for which a finite part of the spectrum of the Dirac operator consists of arbitrarily prescribed eigenvalues with multiplicity 1.