bluuebunny/crossref_metadata_2025 huggingface dataset:
A geometric setting for the quantum planar n-body system, and a U(n−1) basis for the internal states
Authors: Iwai Toshihiro
Published: July 2002

In a previous paper [J. Math. Phys. 28, 964 (1987)], the author showed that the internal motion of a molecule, a many-body system in the Born–Oppenheimer approximation, can be well described in terms of the gauge theory or of the connection theory in differential geometry. However, the scope of that paper centers on the planar triatomic molecule in order to put forward the gauge theory in an explicit manner. This paper is a continuation of the previous one and gives the generalization to the planar multiatomic molecule. The internal space of the n-atomic molecule proves to be diffeomorphic to R+×CPn−2, the product of the positive real numbers and the complex projective space. The internal states of the molecule are described as cross sections in complex line bundles over the internal space. Introduction of the complex line bundles is a geometric consequence of the angular momentum conservation law, because cross sections in each complex line bundle are in one-to-one correspondence with eigenstates that have a fixed total angular momentum eigenvalue. The internal Hamiltonian operator is obtained, which acts on the cross sections in the complex line bundle. Further, boson calculus is performed to obtain a complete basis of internal states of the molecule, using the harmonic oscillator annihilation and creation operators. As a result, carrier spaces of unitary irreducible representations of the unitary group U(n−1), which are characterized by two integers, are realized as finite-dimensional subspaces of the space of the square integrable cross sections in the complex line bundle.
Generalized eigenfunction expansions for scattering in inhomogeneous three-dimensional media
Authors: Rose James H. , Cheney Margaret
Published: July 2002

An infinite number of ways are developed for representing a function in terms of the (generalized) eigenfunctions of a three-dimensional scattering problem and simple known auxiliary functions. The freedom represented by this variety of expansions arises from the causal nature of the wave equations considered. The new expansions are shown to generalize both the Fourier and Radon transforms. An application of the new expansions to the inverse scattering problem is given. It is shown (under some restrictions) that the scattering amplitude and potential are related via one of the generalized transforms.
Hauser–Malhiot spaces admitting a perfect fluid energy-momentum tensor
Authors: Papacostas Taxiarchis
Published: July 2002

In this paper a special case of Hauser–Malliot (HM) space-times is examined in the presence of a perfect fluid source. The obtained solutions are all known except for a generalization of a stationary axisymmetric solution found by Kramer [Class. Quantum Gravit. 1, L3 (1984)].
On the quantization of Poincaré and de Sitter gauge models
Authors: Aldrovandi R. , Pereira J. G.
Published: July 2002

The gauge model based on the Yang–Mills equations for the Poincaré group cannot be consistently quantized, at least in a perturbative approach. The regulated theory, obtained by adding the counterterms required by consistency and renormalizability, is just the gauge theory for a de Sitter group.
Loop algebras and superalgebras based on S 7
Authors: Englert F. , Sevrin A. , Troost W. , Van Proeyen A. , Spindel Ph.
Published: July 2002

Loop algebras of parallel transformations are defined on seven spheres. These algebras can be extended with eight fermionic charges and the Virasoro algebra to a soft superalgebra using the geometry of cosets of supergroups, or to a nonassociative infinite-dimensional N=8 d=2 superconformal algebra.
Unitary lowest weight representations of the noncompact supergroup OSp(2n/2m,R)
Authors: Günaydin M. , Hyun S. J.
Published: July 2002

The oscillator construction of the unitary irreducible lowest (highest) weight representations of the noncompact supergroup OSp(2n/2m,R) with even subgroup SO(2n)×Sp(2m,R) is given. In particular, simple rules for determining the SO(2n)×Sp(2m,R) decomposition of the unitary lowest weight representations of OSp(2n/2m,R) are derived.
Finitely many sphere interactions in quantum mechanics: Nonseparated boundary conditions
Authors: Dabrowski L. , Shabani J.
Published: July 2002

Let Ḣ be the closure of the restriction of the three-dimensional Laplacian −Δ on the domain C∞0(R3\Σ), where Σ=∪Nj=1∂K(0,Rj) and ∼(K(0,Rj)) is a closed ball of radius Rj centered at the origin in R3. It is well known that Ḣ is a closed symmetric operator with deficiency indices (∞,∞). In this paper all self-adjoint (s.a.) extensions of Ḣ are constructed; these extensions contain as particular cases the quantum Hamiltonian describing concentric δ- and δ′-sphere interactions. It is also shown that the s.a. extensions of Ḣ may be obtained as norm-resolvent limits of momentum cutoff and scaled separable potentials.
On quantum integrable models related to nonlinear quantum optics. An algebraic Bethe ansatz approach
Authors: Jurčo Branislav
Published: July 2002

A unified approach based on Bethe ansatz in a large variety of integrable models in quantum optics is given. Second harmonics generation, three-boson interaction, the Dicke model, and some cases of four-boson interaction as special cases of su(2)⊕su(1,1)-Gaudin models are included.
A Rodrigues formula approach to determining closed-form solutions to the Schrödinger equation for symmetric anharmonic oscillators
Authors: Leach P. G. L. , Flessas G. P. , Gorringe V. M.
Published: July 2002

A natural generalization to the Rodrigues formula for the Hermite polynomials leads to the definition of sets of polynomials. Under certain constraints they are used to explain the existence of closed-form solutions to the Schrödinger equation for symmetric anharmonic potential wells and to obtain further closed-form solutions.
Algebras connected with the Z
n elliptic solution of the Yang–Baxter equation
Authors: Bo-yu Hou , Hua Wei
Published: July 2002

The quantum and classical algebras connected with the Zn-symmetric elliptic solution of the Yang–Baxter equation are derived; their structure constants and the relations between the quantum algebra and the classical one are investigated in detail. Moreover, the trigonometric limit of these algebras is worked out.
The generalized continued fractions and potentials of the Lennard-Jones type
Authors: Znojil Miloslav
Published: July 2002

For a broad class of the strongly singular potentials V(r), which are defined as superpositions of separate power-law components, the general solution of the corresponding Schrödinger differential equation is constructed as an analog of Mathieu functions. The analogy is supported by the use of the (generalized) continued fractions. The questions of convergence are analyzed in detail.
Extension of two-body Dirac equations to general covariant interactions through a hyperbolic transformation
Authors: Crater H. W. , Van Alstine P.
Published: July 2002

In previous work, Dirac’s constraint mechanics and supersymmetry were used to construct two-body Dirac equations for particles interacting through world scalar and vector potentials. The resulting compatible 16-component wave equations, 𝒮1ψ=γ51(γ1⋅(p1−Ã1) +m1+S̃1)ψ=0, 𝒮2ψ=γ52(γ2⋅(p2−Ã2) +m2+S̃2)ψ=0, yield Schrödinger-like equations in the center-of-momentum system with simpler structure than that possessed by the Bethe–Salpeter equation or any of its standard three-dimensional truncations. For strong interactions, these equations have yielded a relativistic quark model for meson spectroscopy, while for electromagnetic interactions they have yielded a family of exact singlet positronium solutions. This paper uncovers a hyperbolic interaction structure in these equations. This structure is used to generalize these equations to include pseudoscalar, pseudovector, and tensor interactions.
On quantum fields that generate local algebras
Authors: Buchholz Detlev
Published: July 2002

The properties of quantum fields generating local nets of von Neumann algebras are discussed. For fields that satisfy certain energy bounds, necessary and sufficient conditions for the existence of such a net and for the existence of product states are given in terms of the vacuum expectation values.
Nontransititive imprimitivity systems for the Galilei group
Authors: Cassinelli Gianni , Lahti Pekka J.
Published: July 2002

A class of nontransitive imprimitivity systems and the corresponding projective unitary representations for the inhomogeneous Galilei group are worked out with the Mackey–Varadarajan method of group representations.
Separation of variables and exact solution to the Dirac equation in curvilinear orthogonal coordinates with cylindrical symmetry
Authors: Villalba Victor M.
Published: July 2002

In the present article, a complete separation of variables in the Dirac equation for a free particle is achieved in parabolic cylinder and elliptical coordinates. The resulting system of ordinary differential equations is solved and the asymptotic behavior of the spinor solution is discussed.
Reflection coefficient beyond all orders for singular problems. I: Separated critical points on the nearest critical level line
Authors: Hu Jishan , Kruskal Martin
Published: July 2002

This paper generalizes the Pokrovskii–Khalatnikov method to calculate the nontrivial behavior of the reflection coefficient for singular problems. Two different classes of reflection coefficient problems are considered.
Bloch-wave scattering by crystal defects
Authors: Newton Roger G.
Published: July 2002

The Korringa, Kohn, and Rostoker (KKR) conditions for Bloch waves that describe the motion of electrons in a perfect crystal, and the scattering of these waves by crystal impurities, are formulated by the methods of time-independent scattering theory without recourse to angular-momentum expansions. Generalized Levinson theorems (Friedel sum rules) are proved, including the possibility of half-bound states.
A nonstandard representation of Feynman’s path integrals
Authors: Nakamura Touru
Published: July 2002

A nonstandard path space is constructed that gives the mathematically rigorous formulation for the path integral representation of the fundamental solution to the Cauchy problem for the Dirac equation in (1+1)-dimensional space-time. Nonstandard analysis makes the mathematical concepts elementary, consequently, the procedures to prove theorems are considerably simplified. A difference scheme with infinitesimal spacing is available in determining the probability distribution over the path-space and this method is also available for the heat equation and for the free Schrödinger equation.
A q-analog of the quantum central limit theorem for SUq(2)
Authors: Lenczewski Romuald , Podgórski Krzysztof
Published: July 2002

A q-analog of the central limit theorem for SUq(2), q≳0, is studied. It is shown that the limits of the moments and q-exponential generating functions for coherent states give for q≳1 (0<q<1) the harmonic q-oscillator (q−1-oscillator) introduced by Biedenharn and Macfarlane.
Homogeneous space-times with seven isometries
Authors: Rebouças M. J. , Teixeira A. F. F.
Published: July 2002

A new homogeneous space-time with seven isometries is derived by means of a complex transformation. It is shown that it is conformally flat and its Weyl-flat form is exhibited. Its Killing vector fields and the corresponding Lie algebra are also presented. An integral representation of these vector fields is found. Sources for the new space-time geometry are obtained by combining Synge’s approach to Einstein’s equations with energy conditions and Segrè classification of the energy-momentum tensors. A comparison with other rare space-times possessing a G7 of motions is made by using a suite of computer algebra programs.
Irreducible representations of braid groups
Authors: Tong Dian-Min , Zhu Cheng-Jiu , Ma Zhong-Qi
Published: July 2002

The irreducible representations of the braid groups, described by the Young patterns, are obtained in terms of a set of commutant operators. The relations between our representations and the monodromy representations based on the minimal representation of the quantum enveloping algebra Uqsl(2) are discussed briefly.
Investigation of Painlevé property under time singularities transformations
Authors: Goriely Alain
Published: July 2002

In this paper, the relation between the Painlevé property for ordinary differential equations and some coordinate transformations in the complex plane is investigated. Two sets of transformations are introduced for which the transformations of singularities structure in the complex plane are explicitly computed. The first set acts only on the dependent variables while the second one acts also on the independent variables. It allows for defining an extended Painlevé test which is coordinates invariant. Our results are applied on various problems arising in the singularity analysis such as the problem of negative resonances and the weak Painlevé conjecture. The method for finding new first integrals for dynamical systems is also shown.
A topological-field-theory approach to the non-Abelian Stokes theorem
Authors: Broda B.
Published: July 2002

A holomorphic path-integral representation for the parallel-transport and holonomy operator is derived. It is shown that a new form of the non-Abelian Stokes theorem follows from the Abelian Stokes theorem used for a topological field theory in an external gauge field.
Symplectic reduction and topology for applications in classical molecular dynamics
Authors: Lin F. J. , Marsden J. E.
Published: July 2002

This paper aims to introduce readers with backgrounds in classical molecular dynamics to some ideas in geometric mechanics that may be useful. This is done through some simple but specific examples: (i) the separation of the rotational and internal energies in an arbitrarily floppy N-body system and (ii) the reduction of the phase space accompanying the change from the laboratory coordinate system to the center of mass coordinate system relevant to molecular collision dynamics. For the case of two-body molecular systems constrained to a plane, symplectic reduction is employed to demonstrate explicitly the separation of translational, rotational, and internal energies and the corresponding reductions of the phase space describing the dynamics for Hamiltonian systems with symmetry. Further, by examining the topology of the energy-momentum map, a unified treatment is presented of the reduction results for the description of (i) the classical dynamics of rotating and vibrating diatomic molecules, which correspond to bound trajectories and (ii) the classical dynamics of atom–atom collisions, which correspond to scattering trajectories. This provides a framework for the treatment of the dynamics of larger N-body systems, including the dynamics of larger rotating and vibrating polyatomic molecular systems and the dynamics of molecule–molecule collisions.
Stochastic quantization for quantum mechanics and conditional probabilities
Authors: Houard Jean-Claude , Irac-Astaud Michèle
Published: July 2002

The applicability of the stochastic quantization method to the quantum mechanics is examined. The imaginary character of the action is taken into account by the introduction of a pseudowhite noise defined by an imaginary Gaussian density. The intervening of end points in the paths contributing to the functional integral is treated by introducing Gaussian densities with vanishing variance. The same method allows the calculation of conditional probabilities for the Brownian motion.
A note on the Lorentz transformations linking initial and final four-vectors
Authors: Ungar Abraham A.
Published: July 2002

The set of all (homogeneous, proper, orthochronous) Lorentz transformations that link two given forward-timelike four-vectors with equal norms, as well as the condition under which a unique Lorentz transformation is singled out, is completely determined, and presented in a form suitable for immediate reduction to Galilean transformations by letting c→∞,c being the speed of light in empty space. Analogies to the intuitively well-understood Galilean transformation group are obvious. Thus, for instance, the ordinary velocity addition operator ‘‘+’’ involved in the determination of a Galilean transformation link becomes ⊕, the relativistic velocity addition operator involved in the determination of a Lorentz transformation link. The analogies shared by Galilean and Lorentz transformation links were overlooked by explorers since, as opposed to the associative–commutative binary operation + in the Euclidean three-space R3, the binary operation ⊕ in R3c = {v∈R3:‖v‖<c} is neither associative nor commutative.
The general solution of the three-dimensional acoustic equation and of Maxwell’s equations in the infinite domain in terms of the asymptotic solution in the wave zone
Authors: Moses Harry E. , Nagem Raymond J. , Sandri Guido v. H.
Published: July 2002

The general solution of the three-dimensional scalar wave equation (or acoustic equation) and of Maxwell’s equations in the infinite spatial domain is given in terms of the asymptotic forms for large times in the future and in the past, or, equivalently, in terms of the fields in the wave zone. One is therby able to obtain the exact solutions from arbitrary solutions in the wave zone. It is shown that the exact fields computed from an arbitrary wave zone solution always satisfy an initial value problem, and that, therefore, they are always physical. In contrast to earlier derivations of related results which required the use of Radon transforms and the introduction of somewhat sophisticated geometrical concepts, the derivations are simple and use only elementary properties of the Fourier transform.
Asymmetric anharmonic oscillators in the Hill-determinant picture
Authors: Znojil Miloslav
Published: July 2002

A new version of the Hill-determinant construction of bound states is proposed. It is based on the matching of two suitable power-series Ansätze for the wave function. For the asymmetric well potential V(x)=ax+bx2+cx3+dx4 the convergence of the approximants to the physical solutions is proved. For b≳bminimal, the Taylor coefficients for this anharmonic oscillator’s wave function are specified by four-term recurrences.
Construction of a complex-valued fractional Brownian motion of order N
Authors: Sainty Philippe
Published: July 2002

In this paper, a Brownian motion of order n (n≳2) is defined by a probabilistic approach different from Hochberg’s and Mandelbrot’s. This process is constructed from sums of independent R+1/n-valued random variables (rv) (where R+1/n={z∈C; zn∈R+}). Many properties of the real standard Brownian motion are generalized at order n, but in the case n≳2, it is interesting to describe the Brownian motion of order n on the σ algebra ⊗[B(R+1/n)] R+ [where B(R+1/n) is the σ algebra generated by sets of type A(0,h)={z∈C; zn∈[0;hn[},(hεR+*)]. This σ algebra is totally different from ⊗[B(R)] R+. Thus this study shows the fractal nature of the Brownian motion of order n, and given invariance scale (self-similarity) properties. Then, a stochastic integral and an Itô–Taylor lemma at order n are given to allow the representation of the solution of the heat equation of order n by a probabilistic average. All these results can be obtained via nonstandard analysis methods (infinitesimal time discretization). Finally, one remarks that this process has a.s (almost surely) continuous sample paths, infinite variance, and independent increments, whereas the fractional Brownian motion of Mandelbrot has a.s continuous sample paths, finite variance, and interdependent increments.
On the phase diagram for a class of continuous systems
Authors: Gielerak Roman
Published: July 2002

Some Gibbsian aspects of the scheme known as the sine-Gordon transformation are discussed. Dobrushin–Lanford–Ruelle (DLR) equilibrium functional equations are introduced and analyzed. Certain uniqueness theorems are proved. Thermodynamical properties of the corresponding Gibbs measures are discussed. In particular, the Gibbs variational principle and the information gain principle are proved.
On the algebra of coupled SO(3) tensors
Authors: Rao K. Srinivasa , Van der Jeugt J. , Vanden Berghe G.
Published: July 2002

The commutation relation for coupled SO(3) tensor operators is constructed. This gives rise to a Lie algebra for which 9j symbols appear in the structure constants. In an example it is shown how an exceptional Lie subalgebra can be realized. The closure requirement for the exceptional Lie algebra does not seem to be related to nontrivial 9j symbols but rather to vanishings of relations involving 6j symbols.
The factorial Schur function
Authors: Chen William Y. C. , Louck James D.
Published: July 2002

The application of the divided difference of a function to the inhomogeneous symmetric functions (factorial Schur functions) of Biedenharn and Louck is shown to lead to new relations and simplified proofs of their properties. These results include determinantal definitions and the factorial Jacobi–Trudi identities with extensions to skew versions. Similar properties of a second class of symmetric functions depending on an arbitrary parameter, and of importance for generalized hypergeometric functions and series, are shown also to be derivable from the divided difference notion, slightly extended.
On the integrability of differential forms related to nonequilibrium entropy and irreversible thermodynamics
Authors: Chen M. , Eu Byung Chan
Published: July 2002

The Boltzmann equation has been used in the literature to show that the entropy differential consists of a Pfaffian form in the space of conserved and nonconserved variables (moments) and a term related to the energy dissipation due to the irreversible processes in the system. The said Pfaffian form is called the compensation differential. In this paper, the integrability of the compensation differential is examined by means of the theory of differential forms. The integrability conditions turn out to be generalized forms of the Maxwell relations in equilibrium thermodynamics. It is also shown that the generalized form of the Gibbs–Duhem relation can be seen as an equivalent of the integrability conditions. This conclusion is drawn by using the notion of homotopy operator. The Caratheodory principle is also applied to make the connection with the second law of thermodynamics more intimate than the direct but mathematically more abstract approach using the integrability conditions. The meaning of the integrating factor is clarified by using the notion of contact temperature since the integrating factor, a mathematical function, is endowed with a thermodynamically operational meaning when it is identified with the inverse local absolute temperature through the notion of contact temperature, and the compensation differential is thereby made a thermodynamically meaningful equation governing nonequilibrium processes. Through this study it is shown that in irreversible thermodynamics the compensation differential can play the role parallel to the equilibrium Gibbs relation for the entropy in equilibrium thermodynamics and thus can serve as the foundation on which to formulate a theory of irreversible processes.
Localization properties of wavelet transforms
Authors: Holschneider Matthias
Published: July 2002

Estimates are given of the localization of wavelet transforms on the half-plane in terms of the localization of the analyzed function in time and frequency. Then a precise formulation of the wavelet analysis of tempered distributions is given. The results in this paper are well known to the wavelet community. However, they are nowhere explicitly written up though they are quite useful. This paper therefore serves as reference for these kinds of results.
The super-rotation Racah–Wigner calculus revisited
Authors: Daumens M. , Minnaert P. , Mozrzymas M. , Toshev S.
Published: July 2002

It is shown that, since the finite dimensional representations of the super-rotation algebra are characterized by the superspin j and the parity λ of the representation space, all features of the Racah–Wigner calculus: Clebsch–Gordan coefficients, recoupling coefficients as well as the Wigner and Racah symbols depend on both j and λ. However, it is noticed that the dependence on the parities of the Wigner and Racah symbols can be factorized out into phases so that one can define parity-independent super S3−j and S6−j symbols. The properties of these symbols are analyzed, in particular, it is shown that the S6−j symbols possess a symmetry similar to the Regge symmetry satisfied by the rotation 6−j symbols. Analytical and numerical tables of the symbols are given for the lowest values of their arguments.
Continuity of the S matrix for the perturbed Hill’s equation
Authors: Clemence Dominic P. , Klaus Martin
Published: July 2002

The behavior of the scattering matrix associated with the perturbed Hill’s equation as the spectral parameter approaches an endpoint of a spectral band is studied. In particular, the continuity of the scattering matrix at the band edges is proven and explicit expressions for the transmission and reflection coefficients at those points are derived. All possible cases are discussed and our fall-off assumptions on the perturbation are weaker than those made by other authors.
Time-dependent Lagrangian systems: A geometric approach to the theory of systems with constraints
Authors: Rañada Manuel F.
Published: July 2002

A geometric approach to the theory of time-dependent regular Lagrangian systems with constraints is presented using the framework of the exact contact manifold (TQ×R,ΘL). The main subject of the article concerns the properties of the time-dependent nonholonomic constraints and the geometric approach to the method of the Lagrange multipliers. It is shown that every constraint determines a contact one-form, a vertical vector field, and a nonvertical vector field. The explicit form of the vector field representing the constrained dynamics is obtained and, finally, the properties of all these one-forms and vector fields are discussed.
A gravitational lens need not produce an odd number of images
Authors: Gottlieb Daniel H.
Published: July 2002

Given any space–time M without singularities and any event O, there is a natural continuous mapping f of a two-dimensional sphere into any spacelike slice T not containing O. The set of future null geodesics (or the set of past null geodesics) form a two-sphere S2 and the map f sends a point in S2 to the point in T which is the intersection of the corresponding geodesic with T. Considering the f for each point of a world-line W gives us a map F:S2×W→T. The local degree of F at a regular value y in T has the same parity as the number of null geodesics from W to y.
Representations of super Yangian
Authors: Zhang R. B.
Published: July 2002

The classification of the finite-dimensional irreducible representations of the super Yangian associated with the Lie superalgebra gl(1‖1) is presented in detail.
Affine Lie algebraic origin of constrained KP hierarchies
Authors: Aratyn H. , Gomes J. F. , Zimerman A. H.
Published: July 2002

An affine sl(n+1) algebraic construction of the basic constrained KP hierarchy is presented. This hierarchy is analyzed using two approaches, namely linear matrix eigenvalue problem on hermitian symmetric space and constrained KP Lax formulation and it is shown that these approaches are equivalent. The model is recognized to be the generalized non-linear Schrödinger (GNLS) hierarchy and it is used as a building block for a new class of constrained KP hierarchies. These constrained KP hierarchies are connected via similarity-Bäcklund transformations and interpolate between GNLS and multi-boson KP-Toda hierarchies. Our construction uncovers the origin of the Toda lattice structure behind the latter hierarchy.
Six-parameter solution of the Einstein–Maxwell equations possessing equatorial symmetry
Authors: Manko V. S. , Martín J. , Ruiz E.
Published: July 2002

The general asymptotically flat two-soliton solution of the Einstein–Maxwell equations symmetric about the equatorial plane is considered herein. The complex potentials of this solution and the corresponding metric functions are obtained in the explicit compact form. The balance of two symmetric charged spinning objects is analyzed.
An infinite sum related to WKB eigenvalues
Authors: Millane R. P.
Published: July 2002

An infinite sum that appears in the error term for variational estimates of eigenvalues of the Helmholtz equation is examined. The sum is evaluated directly, confirming a result based on a higher-order variational analysis. Use of this method for evaluating related infinite sums is discussed.
An integral equation of Muskhelishvili type: Strong quantum electrodynamics
Authors: Atkinson D. , Gusynin V. P.
Published: July 2002

An equation that arises out of the bifurcation analysis of an improvement of the nonperturbative equations for the electron mass function in quenched quantum electrodynamics is analyzed. In the quasilinear approximation, the integral equation is solved by Mellin transformation, followed by the calculation of the Muskhelishvili index of the resultant singular integral operator.
Transformation theory of the q-oscillator
Authors: Finkelstein Robert , Marcus Eran
Published: July 2002

A deformation of quantum mechanics obtained by replacing commutators by q-commutators (where q is close enough to unity to be compatible with experiment) is studied. This idea is explored by studying the q-harmonic oscillator in both configuration space and momentum space. The complete set of states in both representations as well as the transformation between x and p space is obtained. The states as well as the matrix elements lie in the SUq(2) algebra. To obtain expectation values and transition probabilities one must average over the SUq(2) algebra.
Exact solutions of Heisenberg equations for multiphoton Jaynes–Cummings model
Authors: Iwasawa H. , Matsuo K.
Published: July 2002

The Heisenberg operator solutions for the multiphoton Jaynes–Cummings model will be presented. By solving the operator differential equations, the exact solutions for the photon annihilation operator a(t), the atomic dipole-moment operator σ−(t), and the population difference operator σz(t) are obtained. The time dynamical behaviors for the operators are compared with the results obtained by the density operator methods.
Invariant spinors and reduced Dirac equations under subgroups of the Euclidean group in four-dimensional Euclidean space
Authors: Legaré M.
Published: July 2002

The most general Dirac spinors globally invariant under the representatives of the conjugacy classes of the connected part to the identity of the Euclidean group in four dimensions are determined. The Dirac equation is reduced under each subgroup representative and a relation to the reduction by symmetry of a linear system of the self-dual Yang–Mills equations is given.
A class of shear-free, non-static models for anisotropic magnetofluid systems
Authors: Duggal K. L. , Asgekar G. G. , Aherkar S. M.
Published: July 2002

A class of spherically symmetric non-static models compatible with relativistic anisotropic magnetofluid systems is obtained under the special feature that flow lines are shear-free but expanding and non-geodesic. In addition, some conformally flat solutions leading to generalized versions of Robertson–Walker models are derived. Moreover, it is proved that if the space–time admits conformal motion then the associated solution does not admit proper conformal symmetry vectors. The models found are of Petrov type I, D, or O.
A completely integrable Hamiltonian system
Authors: Calogero F. , Françoise J.-P.
Published: July 2002

The dynamical system characterized by the Hamiltonian H(q,p)=∑j,k=1n pjpkf(q dj−qk) with f(x)=λ+μ cos(νx)+μ′ sin(ν‖x‖) is completely integrable. Here n is an arbitrary positive integer and λ,μ,μ′,ν are 4 arbitrary constants (λ and μ real, μ′ and ν both real or both imaginary).
On curvature in noncommutative geometry
Authors: Dubois-Violette M. , Madore J. , Masson T. , Mourad J.
Published: July 2002

A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.
Differential calculi and linear connections
Authors: Dimakis Aristophanes , Madore J.
Published: July 2002

A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a one-to-one correspondence, between the module structure of the 1-forms and the metric torsion-free connections on it. In the commutative limit the connection remains as a shadow of the algebraic structure of the 1-forms.
Noncommutative structure of singularities in general relativity
Authors: Heller Michael , Sasin Wiesl/aw
Published: July 2002

Initial and final singularities in the closed Friedman world model are typical examples of malicious singularities. They form the single point of Schmidt’s b-boundary of this model and are not Hausdorff separated from the rest of space–time. The method of noncommutative geometry, developed by A. Connes and his co-workers, is applied to this case. We rephrase Schmidt’s construction in terms of the groupoid Ḡ of orthonormal frames over space–time and carry out the ‘‘desingularization’’ process. We define the line bundle τ:Ω1/2→Ḡ over Ḡ and change the space of its cross sections into an involutive algebra. This algebra is represented in the space of operators on a Hilbert space and, with the norm inherited from these operators, it becomes a C*-algebra. The initial and final singularities of the closed Friedman model are given by two distinct representations of this C*-algebra in the space of operators acting on the Hilbert space L2(O(3,1)).
The eigenvalues of the Laplacian on a sphere with boundary conditions specified on a segment of a great circle
Authors: Abawi Ahmad T. , Dashen Roger F. , Levine Herbert
Published: July 2002

We prove that the eigenvalues of the Laplacian on a sphere with a Dirichlet boundary condition specified on a segment of a great circle lie between an integer and a half-integer and for a Neumann boundary condition they lie between a half integer and an integer. These eigenvalues correspond to the eigenvalues of the angular part of the Laplacian with boundary conditions specified on a plane angular sector, which are relevant in the calculation of scattering amplitude. These eigenvalues can also be used to determine the behavior of the fields near the tip of a plane angular sector as a function of the distance to the tip. The first few eigenvalues for both Dirichlet and Neumann boundary conditions are calculated. The same eigenvalues are also calculated using the Wentzel–Kramers–Brillouin (WKB) method. There is excellent agreement between the exact and the WKB eigenvalues.
Global existence and exponential decay for hyperbolic dissipative relativistic fluid theories
Authors: Kreiss Heinz-Otto , Nagy Gabriel B. , Ortiz Omar E. , Reula Oscar A.
Published: July 2002

We consider dissipative relativistic fluid theories on a fixed flat, globally hyperbolic, Lorentzian manifold (R×T3,gab). We prove that for all initial data in a small enough neighborhood of the constant equilibrium states (in an appropriate Sobolev norm), the solutions evolve smoothly in time forever and decay exponentially to some, in general undetermined, constant equilibrium state. To prove this, three conditions are imposed on these theories. The first condition requires the system of equations to be symmetric hyperbolic, a fundamental requisite to have a well posed and physically consistent initial value formulation. For the flat space-times considered here the equilibrium states are constant, which is used in the proof. The second condition is a generic consequence of the entropy law, and is imposed on the non-principal part of the equations. The third condition is imposed on the principal part of the equations and it implies that the dissipation affects all the fields of the theory. With these requirements we prove that all the eigenvalues of the symbol associated to the system of equations of the fluid theory have strictly negative real parts, which, in fact, is an alternative characterization for the theory to be totally dissipative. Once this result has been obtained, a straightforward application of a general stability theorem due to Kreiss, Ortiz, and Reula implies the results mentioned above.
Fluctuating metrics in one-dimensional manifolds
Authors: Mendoza Ramón , Gómez Pedro , Moraes Fernando
Published: July 2002

In this work we address the statistical mechanics of fluctuating metrics in two, simple, one-dimensional, manifolds: the unit interval and the unit circle. Faddeev–Popov and zeta-function regularization are used to compute explicitly the partition function without the need of any extra fields in the case of the interval. The addition of bosonic fields to the fluctuating metric background is necessary in order to accomplish complete regularization in the case of the circle.
Integrating singular functions on the sphere
Authors: Dickenstein Alicia , Iriondo Mirta Susana , Rojas Teresita Alejandra
Published: July 2002

We obtain rigorous results concerning the evaluation of integrals on the two-sphere using complex methods. It is shown that for regular as well as singular functions which admit poles, the integral can be reduced to the calculation of residues through a limiting procedure.
Tensor categories of Coxeter type B and QFT on the half plane
Authors: Häring-Oldenburg Reinhard
Published: July 2002

We introduce braided tensor categories which are associated to the braid group of Coxeter type B. Connected by a reconstruction theorem there is the notion of B-braided Hopf algebras. These structures show up in quantum field theories defined on the half plane or on the cylinder.
On the algebraic structure of differential calculus on quantum groups
Authors: Radko O. V. , Vladimirov A. A.
Published: July 2002

Intrinsic Hopf algebra structure of the Woronowicz differential complex is shown to generate quite naturally a bicovariant algebra of four basic objects within a differential calculus on quantum groups—coordinate functions, differential 1-forms, Lie derivatives, and inner derivations—as the cross-product algebra of two mutually dual graded Hopf algebras. This construction, properly taking into account Hopf-algebraic properties of Woronowicz’s bicovariant calculus, provides a direct proof of the Cartan identity and of many other useful relations. A detailed comparison with other approaches is also given.
Exact solution of a boundary value problem in semiconductor kinetic theory
Authors: de Groot E. H. , Dalitz C.
Published: July 2002

An explicit solution of the stationary one-dimensional half-space boundary value problem for the linear Boltzmann equation is presented in the presence of an arbitrarily high constant external field. The collision kernel is assumed to be separable, which is also known as “relaxation time approximation;” the relaxation time may depend on the electron velocity. Our method consists in a transformation of the half-space problem into a non-normal singular integral equation, which has an explicit solution.
Null surfaces and the Bach equations
Authors: Iriondo Mirta , Kozameh Carlos N. , Rojas Alejandra
Published: July 2002

It is shown that the integrability conditions that arise in the null surface formulation (NSF) of general relativity (GR) impose a field equation on the local null surfaces which is equivalent to the vanishing of the Bach tensor. This field equation is written explicitly to second order in a perturbation expansion. The field equation is further simplified if asymptotic flatness is imposed on the underlying space–time. The resulting equation determines the global null surfaces of asymptotically flat, radiative space–times. It is also shown that the source term of this equation is constructed from the free Bondi data at ℐ. Possible generalizations of this field equation are analyzed. In particular we include other field equations for surfaces that have already appeared in the literature which coincide with ours at a linear level. We find that the other equations do not yield null surfaces for GR.
Integrable coupled KdV systems
Authors: Gürses Metin , Karasu Atalay
Published: July 2002

We give the conditions for a system of N-coupled Korteweg de Vries (KdV) type of equations to be integrable. We find the recursion operators of each subclass and give all examples for N=2.
Point vortices on a sphere: Stability of relative equilibria
Authors: Pekarsky Sergey , Marsden Jerrold E.
Published: July 2002

In this paper we analyze the dynamics of N point vortices moving on a sphere from the point of view of geometric mechanics. The formalism is developed for the general case of N vortices, and the details are worked out for the (integrable) case of three vortices. The system under consideration is SO(3) invariant; the associated momentum map generated by this SO(3) symmetry is equivariant and corresponds to the moment of vorticity. Poisson reduction corresponding to this symmetry is performed; the quotient space is constructed and its Poisson bracket structure and symplectic leaves are found explicitly. The stability of relative equilibria is analyzed by the energy–momentum method. Explicit criteria for stability of different configurations with generic and nongeneric momenta are obtained. In each case a group of transformations is specified, modulo which one has stability in the original (unreduced) phase space. Special attention is given to the distinction between the cases when the relative equilibrium is a nongreat circle equilateral triangle and when the vortices line up on a great circle.
On separable Schrödinger–Maxwell equations
Authors: Zhdanov Renat , Lutfullin Maxim
Published: July 2002

We obtain the most general time-dependent potential V(t,x1,x2) enabling separating of variables in the (1+2)-dimensional Schrödinger equation. With the use of this result the four classes of separable Schrödinger–Maxwell equations are constructed.
Picard–Fuchs ordinary differential systems in N=2 supersymmetric Yang–Mills theories
Authors: Ohta Yűji
Published: July 2002

In general, Picard–Fuchs systems in N=2 supersymmetric Yang–Mills theories are realized as a set of simultaneous partial differential equations. However, if the quantum chromodynamics (QCD) scale parameter is used as a unique independent variable instead of moduli, the resulting Picard–Fuchs systems are represented by a single ordinary differential equation (ODE) whose order coincides with the total number of independent periods. This paper discusses some properties of these Picard–Fuchs ODEs. In contrast with the usual Picard–Fuchs systems written in terms of moduli derivatives, there exists a Wronskian for this ordinary differential system and this Wronskian produces a new relation among periods, moduli, and QCD scale parameter, which in the case of SU(2) is reminiscent of the scaling relation of prepotential. On the other hand, in the case of the SU(3) theory, there are two kinds of ordinary differential equations, one of which is the equation directly constructed from periods and the other is derived from the SU(3) Picard–Fuchs equations in moduli derivatives identified with Appell’s F4 hypergeometric system, i.e., Burchnall’s fifth-order ordinary differential equation published in 1942. It is shown that four of the five independent solutions to the latter equation actually correspond to the four periods in the SU(3) gauge theory and the closed form of the remaining one is established by the SU(3) Picard–Fuchs ODE. The formula for this fifth solution is a new one.
Another proof of the triple sum formula for Wigner 9j-symbols
Authors: Rosengren Hjalmar
Published: July 2002

We give a simple proof of the Ališauskas–Jucys triple sum formula for Wigner 9j-symbols.
Kählerian supermanifolds
Authors: Varsaie S.
Published: July 2002

In this paper the concepts of almost Hermitian and Kählerian supermanifolds are introduced. Using the classical Newlander–Nirenberg theorm, it is shown that an almost complex structure J is integrable if it is compatible with ΔL, the supermanifold analog of the Levi-Civita connection. It is proved that an almost Hermitian supermanifold is Kählerian if and only if ΔL is compatible with J.
On the geodesic form of second-order dynamic equations
Authors: Mangiarotti L. , Sardanashvily G.
Published: July 2002

It is shown that any second-order dynamic equation on a configuration bundle Q→R of nonrelativistic mechanics is equivalent to a geodesic equation with respect to a (nonlinear) connection on the tangent bundle TQ→Q. The case of quadratic dynamic equations is analyzed in detail. The equation for Jacobi vector fields is constructed and investigated by the geometric methods.
Vertex operators of Uq[gl(N∧|N)] and highest weight representations of Uq[gl(2|∧2)]
Authors: Yang Wen-Li , Zhang Yao-Zhong
Published: July 2002

We determine the exchange relations of the level-one q-vertex operators of the quantum affine superalgebra Uq[gl(N|∧N)]. We study in detail the level-one irreducible highest weight representations of Uq[gl(2|∧2)], and compute the characters and supercharacters associated with these irreducible modules.
Structural phase transformation and limited miscibility in Jahn–Teller systems
Authors: Ivanov M. A. , Tkachev N. K. , Fishman A. Ya.
Published: July 2002

Immiscibility of crystal systems with Jahn–Teller (JT) ions is studied at low temperatures. Phase transformations under which concentrated JT systems are separated into phases with a high and low concentration of JT ions with structural JT transformations are analyzed. A dependence of the phase diagram topology on the ratio between the cooperative interaction energy in the JT subsystem and the intensity of random crystal fields acting on the JT ions is established. The possibility of low-temperature three-phase equilibrium of two cubic and a tetragonal phase with different concentrations of JT centers is demonstrated for a high degree of substitutional-type disorder.
On the structure of nonlinear waves in liquids with gas bubbles
Authors: Beylich Alfred E. , Gülhan Ali
Published: July 2002

Transient wave phenomena in two-phase mixtures with a liquid as the matrix and gas bubbles as the dispersed phase have been studied in a shock tube using glycerine as the liquid and He, N2, and SF6 as gases having a large variation in the ratio of specific heats and the thermal diffusivity. Two different sizes of bubble radii have been produced , R0=1.15 and 1.6 mm, with a dispersion in size of less than 5%. The void fraction was varied over one order of magnitude, φ0=0.2%–2%. The measured pressure profiles were averaged by superimposing many shots, typically 20. Speeds and profiles were measured for shock waves and for wave packets. Investigation of the wave structure allows one to approach the fundamental question of how the physics on the level of the microstructure influences the behavior on the macroscale. In the theoretical work, modeling on the basis of a hierarchy of characteristic length scales is developed. Bubble interactions, transient heat transfer, and dissipation due to molecular and bulk viscosities are included. Solutions for small void fractions and moderate amplitudes are obtained for the steady cases of shock waves and solitons and are compared with the experimental results.