Selberg trace formula for bordered Riemann surfaces: Hyperbolic,elliptic and parabolic conjugacy classes,and determinants of Maass-Laplacians

The Selberg trace formula for automorphic forms of weightm, on bordered Riemann surfaces is developed. The trace formula is formulated for arbitrary Fuchsian groups of the first kind with reflection symmetry which include hyperbolic, elliptic and parabolic conjugacy classes. In the case of compact bordered Riemann surfaces we can explicitly evaluate determinants of Maass-Laplacians for both Dirichlet and Neumann boundary-conditions, respectively. Some implications for the open bosonic string theory are mentioned.Address from August 1993: II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany     

Agostic deformations based on electron delocalization in the alkyllithium-complex [(2-(Me3Si)2CLiC5H4N)2

Topological analysis of experimental and theoretical charge densities in the title complex [(2-(Me3Si)2CLiC5H4N)2] 1 reveals the nature of the agostic deformation postulated for this complex: delocalization of the Li-C bonding electrons over the entire agostic alkyl group controls the formation of an acute Li-C-Si angle and thus a sufficient electronic saturation of the electron deficient lithium atom via secondary interactions.     

Selberg super-trace formula for super Riemann surfaces II: Elliptic and parabolic conjugacy classes,and Selberg super-zeta functions

Further contributions developing a super analogue of the classical Selberg trace formula, the Selberg super-trace formula, are presented. This paper deals with the calculation of contributions arising from elliptic and parabolic conjugacy classes to the Selberg super-trace formula for super Riemann surfaces. Analytic properties and the functional equation for the corresponding Selberg super-zeta functionR ₀,R ₁ andZ _S , respectively, are derived and discussed. In particular, the elliptic contributions to a super Fuchsian group only alter the multiplicities of the trivial zeros and poles of the Selberg super-zeta functionR ₀,R ₁ andZ _S , respectively, already due to the hyperbolic conjugacy classes. The parabolic conjugacy classes introduce new features in the analytical structure.     

Pressure Study on New Superconducting Germanium Clathrates

We investigate the low temperature properties of the recently discovered clathrates Ba 6 Ge 25 and Na 2 Ba 4 Ge 25 by tuning both materials with hydrostatic pressure. At ambient pressure, Ba 6 Ge 25 undergoes a two-step structural phase transition between 230 K and 180 K from metallic behavior to a high-resistivity state. A superconducting transition occurs at T_{C}\approx 0.24\,\hbox{K} out of the resulting bad metal ( \rho_{0}\approx 1.5\,\hbox{m}\Omega\;\hbox{cm} ). With increasing pressure, the structural phase transition is shifted to lower temperature but T C increases drastically. T C reaches a maximum value of 3.85 K at the critical pressure p_{C}\approx 2.8\,\hbox{GPa} , where the structural distortion is completely suppressed and the system exhibits metallic behavior. On replacing 1/3 of the Ba atoms with Na (Na 2 Ba 4 Ge 25 ), no structural transformation is observed below room temperature, and the superconducting transition temperature is higher (T_{C}(p = 0) \approx 1.05\,\hbox{K}) than in the undoped case at ambient pressure but decreases slightly with increasing pressure.     

Coulomb Potentials by Path Integration

The path integral for the potential is evaluated in three different coordinate systems, i. e. in cartesian coordinates, in polar coordinates and in parabolic coordinates. The equivalence between the different approaches is shown and in each case the energyspectrum and the wave functions are explicitly calculated. Furthermore we discuss special cases, including the ring-potential, the Coulomb potential with an Aharonov-Bohm solenoid, and the genuine Coulomb problem. We also point out the separability of the path integral formulation of the ring-potential in spheroidal coordinates, and of the Coulomb potential in spheroidal and spheroconical coordinates, respectively.     

Subhertz-linewidth infrared frequency source with a long-term instability below 5 × 10−15

Distributing a stable, absolute optical reference frequency via fiber network would serve research and development in academia and industry. Lasers stabilized to high-finesse Fabry–Pérot cavities can achieve fractional frequency instabilities of less than 10^?15 for periods up to several seconds. Their instabilities increase for longer averaging times due to a variable frequency drift, with a linear drift component of the order of 10…100 mHz/s. Hydrogen masers, on the other hand, yield an instability floor of a few parts in 10^?15, but suffer from poor stabilities on short timescales. We demonstrate an infrared optical frequency source that combines a cavity-stabilized laser with a hydrogen maser to achieve a residual fractional frequency instability better than 5 × 10^?15 for all averaging times from 0.4 up to 10,000 s. The frequency drift of the system over a period of 40,000 s is less than 30 µHz/s. For obtaining absolute frequency accuracy, the hydrogen maser is referenced to a primary frequency standard.     

Path integration on Darboux spaces

In this paper, the Feynman path integral technique is applied to two-dimensional spaces of nonconstant curvature: these spaces are called Darboux spaces D _I-D _IV. We start each consideration in terms of the metric and then analyze the quantum theory in the separable coordinate systems. The path integral in each case is formulated and then solved in the majority of cases; the exceptions being the quartic oscillators where no closed solution is known. The required ingredients are the path integral solutions of the linear potential, the harmonic oscillator, the radial harmonic oscillator, the modified Pöschl-Teller potential, and the spheroidal wave functions. The basic path integral solutions, which appear here in a complicated way, have been developed in recent work and are known. The final solutions are represented in terms of the corresponding Green’s functions and the expansions into the wave functions. We also sketch some limiting cases of the Darboux spaces, where spaces of constant negative and zero curvature emerge.     

Characterization of synthetic polyelectrolytes by capillary electrophoresis

Capillary electrophoresis in entangled polymer solutions was applied to determine the molecular mass and polydispersity of polyelectrolytes. The separation selectivities of different polyethylene glycols as buffer additive can be correlated to their average molecular mass. A universal curve correlating the selectivity and the molecular mass could be obtained by using the instrinsic viscosity of the polyethylene glycol. The separation of poly(2-vinylpyridine) standards was compared to the separation of poly(4-vinylpyridine) standards. An indirect detection system was developed to characterize the cationic polyelectrolyte polydiallyldimethyl ammonium chloride. Various polymers with oppositely charged groups (polycarboxybetaines) were investigated with respect to structure dependence, pH dependence and molecular mass dependence of interand intramolecular association.