Path integral approach for superintegrable potentials on spaces of non-constant curvature: II. Darboux spaces <Emphasis Type="Italic">D</Emphasis><Subscript>III</Subscript> and <Emphasis Type="Italic">D</Emphasis><Subscript>IV</Subscript>

This is the second paper on the path integral approach of superintegrable systems on Darboux spaces, spaces of non-constant curvature. We analyze five and four superintegrable potentials in the spaces D _III and D _IV, respectively; these potentials were first given by Kalnins et al. We are able to evaluate the path integral in most of the separating coordinate systems, leading to expressions for the Green’s functions, the discrete and continuous wavefunctions, and the discrete energy spectra. In some cases, however, the discrete spectrum cannot be stated explicitly because it is determined by a higher-order polynomial equation. We also show that the free motion in a Darboux space of type III can contain bound states, provided the boundary conditions are appropriate. We can state the corresponding energy spectrum and the wavefunctions. The text was submitted by the authors in English.     

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.     

Path integral for Koenigs spaces

I discuss a path-integral approach for the quantum motion on two-dimensional spaces according to Koenigs, for short “Koenigs spaces”. Their construction is simple: one takes a Hamiltonian from a two-dimensional flat space and divides it by a two-dimensional superintegrable potential. These superintegrable potentials are the isotropic singular oscillator, the Holt potential, and the Coulomb potential. In all cases, a nontrivial space of nonconstant curvature is generated. We can study free motion and the motion with an additional superintegrable potential. For possible bound-state solutions, we find in all three cases an equation of the eighth order in the energy E. The special cases of the Darboux spaces are easily recovered by choosing the parameters accordingly. The text was submitted by the authors in English.     

<Emphasis Type="Italic">D</Emphasis>-Branes in the Euclidean <Emphasis Type="Italic">AdS</Emphasis><Subscript>3</Subscript> and <Emphasis Type="Italic">T</Emphasis>-Duality

We show that D-branes in the Euclidean AdS ₃ can be naturally associated to the maximally isotropic subgroups of the Lu–Weinstein double of SU(2). This picture makes very transparent the residual loop group symmetry of the D-brane configurations and gives also immediately the D-branes shapes and the σ-model boundary conditions in the de Sitter T-dual of the SL(2,C)/SU(2) WZW model.     

Nondegenerate superintegrable systems in <Emphasis Type="Italic">n</Emphasis>-dimensional Euclidean spaces

We analyze the concept of a nondegenerate superintegrable system in n-dimensional Euclidean space. Attached to this idea is the notion that every such system affords a separation of variables in one of the various types of generic elliptical coordinates that are possible in complex Euclidean space. An analysis of how these coordinates are arrived at in terms of their expression in terms of Cartesian coordinates is presented in detail. The use of well-defined limiting processes illustrates just how all these systems can be obtained from the most general nondegenerate superintegrable system in n-dimensional Euclidean space. Two examples help with the understanding of how the general results are obtained. The text was submitted by the authors in English.     

On the Dirac eigenvalues as observables of the on-shell <Emphasis Type="Italic">N</Emphasis> = 2<Emphasis Type="Italic">D</Emphasis> = 4 Euclidean supergravity

We generalize previous works on the Dirac eigenvalues as dynamical variables of Euclidean gravity and N =1 D = 4 supergravity to on-shell N = 2 D = 4 Euclidean supergravity. The covariant phase space of the theory is defined as the space of the solutions of the equations of motion modulo the on-shell gauge transformations. In this space we define the Poisson brackets and compute their value for the Dirac eigenvalues.      

Path integral approach for spaces of nonconstant curvature in three dimensions

In this contribution, I show that it is possible to construct three-dimensional spaces of nonconstant curvature, i.e., three-dimensional Darboux spaces. Two-dimensional Darboux spaces have been introduced by Kalnins et al., with a path integral approach by the present author. In comparison to two dimensions, in three dimensions it is necessary to add a curvature term in the Lagrangian in order that the quantum motion can be properly defined. Once this is done, it turns out that, in the two three-dimensional Darboux spaces which are discussed in this paper, the quantum motion is similar to the two-dimensional case. In D _3d-I, we find seven coordinate systems which separate the Schrödinger equation. For the second space, D _3d-II, all coordinate systems of flat three-dimensional Euclidean space which separate the Schrödinger equation also separate the Schrödinger equation in D _3d-II. I solve the path integral on D _3d-I in the (u, v, w) system and on D _3d-II in the (u, v, w) system and in spherical coordinates.     

Static and dynamic properties of KCN<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>Cl<Subscript>1−<Emphasis Type="Italic">x</Emphasis></Subscript>

An extended three-body force shell model (ETSM) has been applied to investigate the static and dynamic properties of KCN _x Cl_1−x for the compositionx = 0.56 and 1.0 at 300 K. The phonon dispersion curves computed by us are compared with the single crystal neutron diffraction data. The unusual features of these curves are the upward curvature seen in some of the acoustic branches. This is a result ofK-dependent softening of the phonon due to translation-rotation coupling. The transverse acoustic branch is more soft near the zone centre.     

Passive Advection and the Degenerate Elliptic Operators <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

 We prove estimates for the stationary state n-point functions at zero molecular diffusivity in the Kraichnan model [13]. This is done by proving upper bounds for the heat kernels and Green's functions of the degenerate elliptic operators M _n that occur in the Hopf equations for the n-point functions. Received: 25 August 2001 / Accepted: 30 September 2002 Published online: 20 January 2003 Communicated by A. Kupiainen     

Superconducting state parameters of La<Subscript>100-<Emphasis Type="Italic">C</Emphasis></Subscript>Ga<Subscript><Emphasis Type="Italic">C</Emphasis></Subscript> binary metallic glasses

The theoretical investigations of the superconducting state parameters (SSP) viz. electron-phonon coupling strength λ, Coulomb pseudopotential μ*, transition temperature T _C , isotope effect exponent α and effective interaction strength N _O V of six binary La_100-C Ga _C (C = 16, 20, 22, 24, 26 and 28 at. %) metallic glasses have been reported using Ashcroft’s empty core (EMC) model potential for the first time. Five local field correction functions proposed by Hartree (H), Taylor (T), Ichimaru-Utsumi (IU), Farid et al. (F) and Sarkar et al. (S) are used in the present investigation to study the screening influence on the aforesaid properties. It is observed that the electron-phonon coupling strength λ and the transition temperature T _C are quite sensitive to the selection of the local field correction functions, whereas the Coulomb pseudopotential μ*, isotope effect exponent α and effective interaction strength N _O V show weak dependences on the local field correction functions. The T _C obtained from H-local field correction function are found in qualitative agreement with available experimental data and show almost linear nature with the concentration (C) of ‘Ga’ element. A linear T _C equation is proposed by fitting the present outcomes for H-local field correction function, which is in conformity with other results for the experimental data. Also, the present results are found to be in qualitative agreement with other such earlier reported data, which confirms the superconducting phase in the metallic glasses.      

A <Emphasis Type="Italic">CR</Emphasis>-manifold in general position as an {<Emphasis Type="Italic">e</Emphasis>}-structure

In the paper, it is proved that a “generic” CR-structure is reduced to an {e}-structure. Using the invariant frames obtained as a result of this reduction (an analog of the Darboux frame of classical differential geometry), we give a simple proof of an analog of the Vitushkin theorem on the extension of a germ for a rather broad class of real submanifolds of complex spaces. Partially financially supported by the RFBR grants nos. 05-01-0981 and NSh-2040.2003.1.     

The determination of <Emphasis Type="Italic">α</Emphasis><Subscript><Emphasis Type="Italic">S</Emphasis></Subscript> from <Emphasis Type="Italic">τ</Emphasis> decays revisited

We revisit the determination of α _S (m _τ ²) using a fit to inclusive τ hadronic spectral moments in light of (1) the recent calculation of the fourth-order perturbative coefficient K ₄ in the expansion of the Adler function, (2) new precision measurements from BABAR of e⁺e⁻ annihilation cross sections, which decrease the uncertainty in the separation of vector and axial-vector spectral functions, and (3) improved results from BABAR and Belle on τ branching fractions involving kaons. We estimate that the fourth-order perturbative prediction reduces the theoretical uncertainty, introduced by the truncation of the series, by 20% with respect to earlier determinations. We discuss to some detail the perturbative prediction of two different methods: fixed-order perturbation theory (FOPT) and contour-improved perturbative theory (CIPT). The corresponding theoretical uncertainties are studied at the τ and Z mass scales. The CIPT method is found to be more stable with respect to the missing higher order contributions and to renormalization scale variations. It is also shown that FOPT suffers from convergence problems along the complex integration contour. Nonperturbative contributions extracted from the most inclusive fit are small, in agreement with earlier determinations. Systematic effects from quark-hadron duality violation are estimated with simple models and found to be within the quoted systematic errors. The fit based on CIPT gives α _S (m _τ ²)=0.344±0.005±0.007, where the first error is experimental and the second theoretical. After evolution to M _Z we obtain α _S (M _Z ²)=0.1212±0.0005±0.0008±0.0005, where the errors are respectively experimental, theoretical and due to the evolution. The result is in agreement with the corresponding N³LO value derived from essentially the Z width in the global electroweak fit. The α _S (M _Z ²) determination from τ decays is the most precise one to date.     

Leptonic constant of the pseudoscalar <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript> meson

We calculate the leptonic constant for the pseudoscalar ground state of the B _c meson in the framework of a QCD-motivated potential model taking into account the two-loop anomalous dimension for the heavy quark current in nonrelativistic QCD as matched with full QCD.     

Weak annihilation and the effective parameters <Emphasis Type="Italic">a</Emphasis><Subscript>1</Subscript> and <Emphasis Type="Italic">a</Emphasis><Subscript>2</Subscript> in non-leptonic <Emphasis Type="Italic">D</Emphasis> decays

Based on SU(3) flavor symmetry, many of the quark-graph amplitudes in two-body non-leptonic decays of charmed mesons can be extracted from experiment, which enable us to see the relevance and importance of weak annihilation topologies and to determine the complex parameters a₁ and a₂ to test the factorization approach. It is found that a ₂ /a ₁ in and can be different by a factor of 2, indicating that non-factorizable corrections to the latter are far more important than the former. The relative phase between a₁ and a₂ is about 150°. Weak annihilation topologies induced by nearby resonances via final-state rescattering can be described in a model-independent manner. Although the W-exchange contribution in decays is dominated by resonant final-state interactions (FSIs), its amplitude in VP decays (V: vector meson, P: pseudoscalar meson) receives little contributions from FSIs in the quark-antiquark resonance formation. As a consequence, the sign flip of the W-exchange amplitude in and decays, which is needed to explain the relatively real decay amplitudes of , remains unexplained. SU(3) symmetry is badly broken in some Cabibbo-suppressed modes and this can be accounted for by the accumulation of some modest SU(3) violation in individual quark-graph amplitudes. Received: 19 July 2002 / Published online: 20 November 2002     

Super <Emphasis Type="Italic">D</Emphasis>-Differentiation for <Emphasis Type="Italic">R</Emphasis><Superscript> ∞ </Superscript>-Supermanifolds

The concept of ‘D-Differentiation’, which, in the context of smooth manifolds, generalises Lie and covariant differentiation, is extended to R ^∞-supermanifolds under the name of ‘Super D-Differentiation’. This is done by defining new (non-linear) mappings, called ‘μ-mappings’ and by relating their non-linearity to the Leibniz rule that a derivation must satisfy when it acts on a tensor product. The resulting axiomatics, which is basis-independent and coordinate-free, is then expressed in a general basis (not necessarily holonomic). Super Lie and Super covariant differentiation are, amongst others, special cases of Super D-Differentiation. In particular, the transformation rules for the connection coefficients and the commutation coefficients of non-holonomic bases are obtained. These special cases are found to be in agreement with the DeWitt Super covariant and Super Lie derivatives.      

Two-Dimensional <Emphasis Type="Italic">D</Emphasis><Superscript>−</Superscript>-States in a Longitudinal Magnetic Field

The quantum-well D(−)-states in the presence of magnetic field longitudinal with respect to the growth axis are considered. Within a model of zero-radius potential, an equation is derived that determines the dependence of the D(−)-state binding energy on the parameters of the potential of the structure, coordinates of the D(−) center, and the magnetic field. The results are compared with the experimental dependence of the D(−)-state binding energy on the magnetic field and the data are shown to be in good agreement with calculations for magnetic fields B < 10 T. A dimension factor is defined in the dependence of the binding energy on coordinates for the 2D → 1D → 0D transition. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 25–29.     

Solution of the Equations of Motion for Einstein’s Field in Fractional <Emphasis Type="Italic">D</Emphasis> Dimensional Space-Time

As a continuation of Sadallah et al. work (M. Sadallah, S. Muslih and D. Baleanu, Equations of motion for Einstein’s field in non-integer dimensional space. Czechoslov. J. Phys. 56:323, 2006), the fractional action function S is given as an integration over fractional spatial dimension D _s and fractional time D _t dimension. The variational principle which minimize S leads to Euler-Lagrange equations of motion in D _s +D _t fractional dimensions. As an example we extend our study to obtain the equations of motion for Einstein’s field in fractional D _s +D _t fractional dimensions of N+1 space-time coordinates. It is shown that the time dependent solutions are single valued for only D _s =4 dimensional space. Also the angular solutions are convergent for any value of D _s .     

Interface <Emphasis Type="Italic">D</Emphasis><Superscript>−</Superscript> complexes in a two-dimensional electron system

The excitation spectrum of a two-dimensional electron system in high-quality AlGaAs/GaAs quantum wells has been studied by Raman scattering. New Raman lines due to the excitation of interface D ⁻ complexes in which two electrons localized in a quantum well are coupled to a charged impurity at the quantum well interface have been identified. The ground state of the interface D ⁻ complexes has been found to change in the transverse magnetic field from spin-singlet to spin-triplet, similar to a change in the ground state of the system of two electrons localized in a harmonic potential.     

Betti Numbers of a Class of Barely <Emphasis Type="Italic">G</Emphasis><Subscript>2</Subscript> Manifolds

We calculate explicitly the Betti numbers of a class of barely G ₂ manifolds - that is, G ₂ manifolds that are realised as a product of a Calabi-Yau manifold and a circle, modulo an involution. The particular class which we consider are those spaces where the Calabi-Yau manifolds are complete intersections of hypersurfaces in products of complex projective spaces from which they inherit all their (1, 1)-cohomology and the involutions are free acting.     

Absorption spectra of thin layers of solid solutions Cs<Subscript>2</Subscript>(Cd<Subscript>1 − <Emphasis Type="Italic">x</Emphasis></Subscript>Zn<Subscript><Emphasis Type="Italic">x</Emphasis></Subscript>)I<Subscript>4</Subscript>

The excitonic absorption spectra of thin films of ferroelectric Cs₂CdI₄ and Cs₂ZnI₄ solid solutions are studied for the first time. It is found that, within the whole range of molar concentrations x, the spectra of Cs₂(Cd_{1 − x} Zn _x )I₄ comprise two bands that originate from the bands of the two compounds. It is shown that the exciton transfer occurs most efficiently between the tetrahedrons CdI₄ and ZnI₄ along the b axis of the crystal. Unusual behavior of the concentration with regards to the maxima of long-wavelength excitonic bands E _m (x) agrees well with the developed theory that takes into account dependence on x of the matrix element of the intertetrahedron exciton transfer and that is similar to the theory of Davydov’s splitting in molecular crystals.