bluuebunny/crossref_metadata_2025 huggingface dataset:
Infinite-dimensional representations of the graded Lie algebra (Sp(4):4). Representation of the para-Bose operators with real order of quantization
Authors: Inaba I. , Maekawa T. , Yamamoto T.
Published: March 2003

Infinite-dimensional representations for the system of the two para-Bose operators, which generate the graded Lie algebra (GLA) (Sp(4):4), are constructed by using the irreducible representations (discrete series) of the Lie algebra Sp(4)≊SO(3,2). It is shown that there are four kinds of the irreducible representations of the GLA’s (Sp(4):4), i.e., cases (I), (II), (III), and (IV). Case (I) is described by the three irreducible representations of Sp(4) and corresponds to the para-Bose quantization with real order of the quantization greater than 1 and case (IV) corresponding to the ordinary Bose quantization by the two irreducible representations of Sp(4). Cases (II) and (III), which are described by the four and two irreducible representations of Sp(4), respectively, and cannot be obtained by the method of Fock space, express the representations of the graded Lie algebra (Sp(4):4).
New Stokes’ line in WKB theory
Authors: Berk H. L. , Nevins William McCay , Roberts K. V.
Published: March 2003

The WKB theory for differential equations of arbitrary order or integral equations in one dimension is investigated. The rules previously stated for the construction of Stokes’ lines for Nth-order differential equations, N⩾3, or integral equations are found to be incomplete because these rules lead to asymptotic forms of the solutions that depend on path. This paradox is resolved by the demonstration that new Stokes’ lines can arise when previously defined Stokes’ lines cross. A new formulation of the WKB problem is given to justify the new Stokes’ lines. With the new Stokes’ lines, the asymptotic forms can be shown to be independent of path. In addition, the WKB eigenvalue problem is formulated, and the global dispersion relation is shown to be a functional of loop integrals of the action.
Properties of three-dimensional Cartesian tensors. III. Concerning tensor averages
Authors: Hollingsworth C. A. , Russ Muchere C.
Published: March 2003

Results from the two earlier papers in this series are used to investigate some average tensorial properties of systems in terms of the tensorial properties of their subsystems. Special attention is given to the Boltzmann distribution in which the orientation energy of a subsystem is assumed to be expressible as a power series in the components of a single vector which is fixed in the laboratory system. The orientation probability density is obtained in general form for subsystems with no symmetry (point group C1), and they are given in more explicit form for a few cases of higher symmetry. General expressions for tensor averages are given, and they are applied to a special case as an example.
The symmetry group of the harmonic oscillator and its reduction
Authors: Iwai Toshihiro
Published: March 2003

The symmetry of the harmonic oscillator is dealt with in the Hamiltonian formalism. Unitary operators representing the symmetry are studied from this point of view. Of additional interest is reduction of the symmetry group SU(4) for the four-dimensional harmonic oscillator. Subspaces are determined from the representation spaces for SU(4) so as to give those for SO(4).
Quantization of the conformal Kepler problem and its application to the hydrogen atom
Authors: Iwai Toshihiro
Published: March 2003

Quantization of the conformal Kepler problem is defined and studied in order that the quantized system, which will be referred to as a conformal hydrogen atom, may associate the harmonic oscillator with the hydrogen atom. The conformal hydrogen atom shares with the harmonic oscillator the eigenspaces of negative energies. The four-dimensional conformal hydrogen atom reduces to the three-dimensional ordinary hydrogen atom. The symmetry group SO(4) of the hydrogen atom is brought out from the symmetry subgroup of the harmonic oscillator. The conformal hydrogen atom gives an example of those quantum systems of which the configuration spaces are curved Riemannian spaces with nonconstant scalar curvatures and of which the Hamiltonian operators depend on the scalar curvatures.
Hypervirial-perturbational treatment of the bounded hydrogen atom
Authors: Fernández Francisco M. , Castro Eduardo A.
Published: March 2003

The hydrogen atom model enclosed inside a container with impenetrable walls is studied by means of a method which combines hypervirial theorems and perturbation theory. From hypervirial relations, very useful recursion relationships for diagonal matrix elements of coordinate powers are given. Numerical as well as analytical results are compared with those given in previous works.
Linear deformation problems for the Ernst equation
Authors: Dodd R. K. , Morris H. C.
Published: March 2003

In the last few years several papers have appeared which give linear deformation problems for the Ernst equation. In this paper we investigate the relationships between the different problems and show that essentially there are two distinct problems up to a gauge transformation corresponding to a finite or infinite dimensional realization of the algebraic prolongation structure associated with the Ernst equation.
Spacetimes admitting a vector field whose inner product with the Riemann tensor is zero
Authors: McIntosh C.B.G. , van Leeuwen E. H.
Published: March 2003

The equation vμRμναβ = 0 arises in various places in general relativity, in particular as the integrability conditions of the equations ℒvg = 2φg, where φ is a constant. These are the equations of a homothetic vector field v, with a zero homothetic bivector (dv = 0) in some space-time with metric g, and Rμναβ are the components of the Riemann tensor of that metric in some frame. In this paper the equation vμ Rμναβ = 0 is examined and the components of the Riemann tensor for the spacetimes which admit nonzero solutions vμ of this equation are given. The Petrov types of the Weyl tensors of these spacetimes are listed and, as a result, a correction is then made to a theorem in a paper by Collinson and Fugère about the Petrov types of spacetimes, which admit the type of separation that they require of the Hamilton–Jacobi equation for these spacetimes.
On a class of exact solutions to the Fokker–Planck equations
Authors: Garrido L. , Masoliver J.
Published: March 2003

In this paper we study under which circumstances there exists a general change of gross variables that transforms any Fokker–Planck equation into another of the Ornstein–Uhlenbeck class that, therefore, has an exact solution. We find that any Fokker–Planck equation will be exactly solvable by means of a change of gross variables if and only if the curvature tensor and the torsion tensor associated with the diffusion is zero and the transformed drift is linear. We apply our criteria to the Kubo and Gompertz models.
Some more developments on operators in Krein space. The exponential map
Authors: Behnam E.
Published: March 2003

This paper generalizes the work of J. L. B. Cooper on symmetric operators in a Hilbert space to Pontrjagin and Krein spaces. Existence (and uniqueness) of a solution for Schrödinger’s equation dψ(t)/dt = iAψ(t), ψ(0) = φ∈Π [ = Π+(+̇)π̄] for the bounded decomposable symmetric operator A (the self-adjoint operator A) is studied. Also, the existence of a solution for (1/i)∂ψ(t)/∂t = A*ψ(t), ψ(0) = φ∈Π, where A is a cross-bounded symmetric operator in Π, is discussed. Finally, existence and uniqueness of the equation (1/i)(∂ψ(t)/∂t) = Aψ(t), ψ(0) = φ∈Πk, where A is a self-adjoint operator in the Pontrjagin space Πk, is studied, and the fact that the maps φ→ψ(t) form a group of unitary operators on Πk is deduced.
A new method for the asymptotic evaluation of a class of path integrals
Authors: Luttinger J. M.
Published: March 2003

A general method of calculating the asymptotic behavior of a class of Wiener path integrals is given. These integrals are averages of functionals of the ’’local time.’’ The technique is essentially a variation on the well-known ’’replica’’ method now widely used in condensed matter physics, combined with the Laplace method for evaluating integrals containing a large parameter. The leading term is given, and from the construction one sees that the error is typically of order 1/t.
On the dynamics of diffusions and the related general electromagnetic potentials
Authors: Morato L. M.
Published: March 2003

In this paper the Nelson’s stochastic mechanics is extended to general diffusion motions. A representation theorem is proved which gives a one-to-one correspondence between solutions of certain Schrödinger equations and diffusion processes satisfying appropriate regularity conditions. Exploiting results of stochastic mechanics on Riemannian manifolds it is shown that the real part of the Schrödinger equations corresponding to the considered diffusions can be interpreted as Newton’s second law where the force is produced by generalized electromagnetic potentials.
Properties of three-dimensional Cartesian tensors. I. Some properties of irreducible tensors
Authors: Russ Muchere C. , Hollingsworth C. A.
Published: March 2003

It is shown that the decomposition of three-dimensional Cartesian tensors into their parts which are irreducible under the rotation group is made conceptually simple by explicitly expressing these parts in an embedded form.
Construction of new integrable Hamiltonians in two degrees of freedom
Authors: Holt Craig R.
Published: March 2003

A new procedure for deriving integrable Hamiltonians and their constants of the motion is introduced. We term this procedure the truncation program. Integrable Hamiltonians occurring in the truncation program possess constants of the motion which are polynomials in a perturbation parameter ε. The relationship between this program and the Whittaker program in two degrees of freedom is discussed. Integrable Hamiltonians occurring in the Whittaker program (a generalization of Whittaker’s work) possess constants of the motion which are polynomials in the momentum coordinates. Many previously known integrable Hamiltonians are derived. A new family of integrable double resonance Hamiltonians and a new family of integrable Hamiltonians of the form (p21+p22)/2+V(q1, q2) are derived.
Dyadic Green functions for the time-dependent wave equation
Authors: Marx Egon , Maystre Daniel
Published: March 2003

The theory of dyadic Green functions for a transient electromagnetic field, which obeys the vector wave equation, is presented within the framework of the theory of distributions. First, the the elementary solution of the scalar wave equation is derived, and then it is used to find the general solution of that equation. After establishing the equivalence between Maxwell’s equations and the time-dependent vector wave equation, the dyadic elementary solution is derived and applied to solve the equation. Further properties of dyadic Green functions for the wave equation are derived within the heuristic approach to the theory of Green functions. The paper includes a collection of formulas from the theory of distributions intended to help readers who are not familiar with the subject.
Single integral equation for wave scattering
Authors: Marx Egon
Published: March 2003

When a wave interacts with an obstacle, the scattered and transmitted fields can be found by solving a system of integral equations for two unknown fields defined on the surface of the body. By choosing a more appropriate unknown function, the system of equations is reduced to a single singular integral equation of the first kind. This reduction is done here for transient and monochromatic waves, for a scalar field that obeys the wave equation, and for electromagnetic fields that obey Maxwell’s equations.
Linear invariants of a time-dependent quantal oscillator
Authors: Vaidyanathan R.
Published: March 2003

Explicitly time-dependent invariants, linear in x and p, are found to simplify the solution of the Schrödinger equation with oscillator-type Hamiltonian. As an example, exact dynamics is obtained of the time-dependent quantal oscillator, with damping and subject to external force. The invariant and hence the solution of the Schrödinger equation involve only the amplitude of the classical damped oscillator. The identity of our solution with a recently obtained result is illustrated. A digressive remark on quadratic invariants and the Dirac operator formalism is added.
ℋℋ spaces with an algebraically degenerate right side
Authors: Plebański J. F. , Torres del Castillo G. F.
Published: March 2003

We find the most general form of the key function for an ℋℋ space whose curvature on the right side is algebraically degenerate.
Static gravitational fields in a general class of scalar–tensor theories
Authors: Dutta Choudhury S. B. , Bhattacharya D.
Published: March 2003

The static field equations are investigated within the framework of a general class of scalar-tensor theories of gravitation proposed by Nordtvedt. In the Brans-Dicke and Barker theories, it is shown that in vacuum g00 is functionally related to the scaler field. Two families of exact solutions are also obtained for a static spherically symmetric gravitational field in Barker theory.
Path integrals and stationary phase approximation for quantum dynamical semigroups. Quadratic systems
Authors: Alicki Robert
Published: March 2003

The solution of the Markovian master equation for the quantum open system of n degrees of freedom is formally written in terms of a path integral and the stationary phase approximation is discussed. The exactly soluble models with generators quadratic in position and momentum operators are investigated and the explicit expressions for the space–time propagators for one-dimensional systems are derived.
On inverse problems for plane-parallel media with nonuniform surface illumination
Authors: Siewert C. E. , Dunn W. L.
Published: March 2003

Elementary considerations are used to solve the inverse problem in linear transport theory for the case of variable illumination over the surface of a plane-parallel layer. The developed formalism yields as a special case the inverse solution for the classical searchlight problem.
A Chern number for gauge fields on R4
Authors: Schlafly R.
Published: March 2003

Let F be the curvature of some connection on some principal bundle over R4. I show that if F decays as fast as (r2lnr)−1 as r tends to infinity, then (1/8π2) ∫ trF2 is an integer. If F decays like r−2, any real value is possible. There is an analogous statement for R2n(n≳2), although it fails for R2.
Nonlinear thermal evolution in an inhomogeneous medium
Authors: Kamin Shoshana , Rosenau Philip
Published: March 2003

Various simple transport models of electron temperature in a confined plasma are reducible to the quasilinear equation ρ(x)ut = [c(x)unx]x +A(x)us, −1<x<1, u(±1) = 0.  u is the temperature, ρ(x) the density, and c = g[ ρ(x)] the density-dependent part of the thermal diffusion.  ρ(x) and c(x) may vanish at the plasma edge, rendering the problem singular. The temporal behavior depends critically on the boundedness of R = ℱ+1−1 c−1(x) dx. If R<∞ then in the absence of heat sources, A≡0, every initially given state u(x, 0) evolves toward an algebraically decaying, universal space- time separable solution. Its existence and uniqueness is proved. The method developed in this work may be used to show the equilibrization of the solution in the presence of a heat source of the form A(x)us, s<n, ρ(x)≳0. On the other hand, if R = ∞ and A = 0 then the system becomes isothermalized: u→ū = ℱ+1−1 u(x, 0) ρ(x) dx/ℱ1−1  ρ(x) dx≳0. In such a case addition of heat sources will cause a thermal explosion.
Spectral theory for the periodic sine-Gordon equation: A concrete viewpoint
Authors: Forest M. Gregory , McLaughlin David W.
Published: March 2003

A summary of the spectral theory for quasiperiodic sine- and sinh-Gordon equations is given. Analogies with whole-line solitons and scattering theory motivates the discussion. The relation between the ingredients in the inverse spectral solution of the periodic sine-Gordon equation and physical characteristics of sine-Gordon waves is emphasized. The explicit topics covered are summarized in the table of contents in the Introduction.
Some algebraic, geometric, and system-theoretic properties of the Special Functions of mathematical physics
Authors: Hermann Robert
Published: March 2003

It is known that many of the Special Functions of mathematical physics appear as matrix elements of Lie group representations. This paper is concerned with a beginning attack on the converse problem, i.e., finding conditions that a given function be a matrix element. The methods used are based on a combination of ideas from system theory, functional analysis, Lie theory, differential algebra, and linear ordinary differential equation theory. A key idea is to attach a symbol as an element of a commutative algebra. In favorable cases, this symbol defines a Riemann surface, and a meromorphic differential form on that surface. The topological and analytical invariants attached to this form play a key role in system theory. The Lie algebras of the groups appear as linear differential operators on this Riemann surface. Finally, it is shown how the Picard-Vessiot-Infeld-Hull theory of factorization of linear differential operators leads to realization of many Special Functions as matrix representations of group representations.
New similarity solutions for the Ernst equations with electromagnetic fields
Authors: Fischer Elliot
Published: March 2003

A new class of exact similarity solutions is found for the Ernst equations with electromagnetic fields. The original coupled nonlinear partial differential equations are reduced to a system of coupled nonlinear ordinary differential equations. The reduced system is solvable in a manner identical to previous similarity solutions found by Kaliappan and Lakshmanan. These solutions may be considered the extension of the Curzon solution (static, uncharged) to the stationary, charged solution.
Quantization as a consequence of the symmetry group: An approach to geometric quantization
Authors: Aldaya Víctor , de Azcárraga José A.
Published: March 2003

A method is proposed to obtain the dynamics of a system which only makes use of the group law. It incorporates many features of the traditional geometric quantization program as well as the possibility of obtaining the classical dynamics: The classical or quantum character of the theory is related to the choice of the group, avoiding thus the need of quantizing preexisting classical systems and providing a group connection between the quantum and classical systems, i.e., the classical limit. The method is applied to the free-particle dynamics and the harmonic oscillator.
Joint distributions, quantum correlations, and commuting observables
Authors: Fine Arthur
Published: March 2003

We provide necessary and sufficient conditions for several observables to have a joint distribution. When applied to the bivalent observables of a quantum correlation experiment, we show that these conditions are equivalent to the Bell inequalities, and also to the existence of deterministic hidden variables. We connect the no-hidden-variables theorem of Kochen and Specker to these conditions for joint distributions. We conclude with a new theorem linking joint distributions and commuting observables, and show how violations of the Bell inequalities correspond to violations of commutativity, as in the theorem.
The quartic anharmonic oscillator in stochastic electrodynamics
Authors: Pesquera L. , Claverie P.
Published: March 2003

The case of a slightly anharmonic oscillator (with a βx4 perturbing potential) is examined in the framework of stochastic electrodynamics (SED) in full detail. We obtain the stationary probability density and the mean energy, which differs from the quantum result at order β2. Using Kubo’s linear response theory we obtain the absorption curve: the maximum absorption frequencies do not coincide with the quantum transition frequencies. From the calculation of the emission energy we show that the ’’radiation balance’’ is not exactly satisfied as soon as β≠0, a property which disagrees with the quantum results. Finally, we discuss the consequences of this lack of radiation balance concerning Kirchhoff’s law.
Properties of Mayer cluster expansion
Authors: Mohanty Udayan
Published: March 2003

The Umbral algebra developed by Rota and his co-workers is used to show that the Mayer cluster expansion of the canonical partition function is related to the Bell polynomials. The algebra is also used to find a representation of the partition function and a rederivation of Mayer’s first theorem. Finally, it is shown that in the ’’tree approximation’’ for the cluster integrals, the summation of Mayer’s expression for the canonical-ensemble partition function for a finite number of particles could be performed using Dénes’ and Rényi’s theorems in graph theory.
Comment on calculations of higher-order tadpoles and triangle vertex integral
Authors: Kalinowski M. W. , Seweryński M. , Szymanowski L.
Published: March 2003

It is found that the calculations of higher-order tadpoles and some integrals associated with the triangle diagram reported by Capper and Leibbrandt are incorrect within their dimensional regularization scheme.
Superfiber bundle structure of gauge theories with Faddeev–Popov fields
Authors: Hoyos J. , Quirós M. , Mittelbrunn J. Ramírez , de Urríes F. J.
Published: March 2003

In this work the mathematical structure of principal superfiber bundle Ps is used to give a geometrical description of gauge theories. The base space of Ps is an S4,2-supermanifold Xs (four commuting and two anticommuting variables), and the structure group an Sn,0 supergroup Gs, where n is the dimension of the gauge group for the classical theory. The body of Ps is the usual principal fiber bundle P of gauge theories. Gauge and Faddeev–Popov fields arise as superfields, components of the connections in Ps, in a local coordinate system. BRS (Becchi, Rouet, and Stora) and anti-BRS transformations are gauge transformations, in Ps, of parameters the ghost and antighost superfields, respectively. In the case of soul-flat connections, which are connections in Ps coming from connections in P, the BRS and anti-BRS transformations are finite translations along the anticommuting directions of Xs, and generate an S0,2-supergroup.
Analytic SU(3) states in a finite subgroup basis
Authors: Desmier P. E. , Sharp R. T. , Patera J.
Published: March 2003

A method is given for deriving branching rules, in the form of generating functions, for the decomposition of representations of SU(3) into representations of its finite subgroups. Interpreted in terms of an integrity basis, the generating functions define analytic polynomial basis states for SU(3), which are adapted to the finite subgroup.
Quantum evolution as a parallel transport
Authors: Asorey M. , Cariñena J. F. , Paramio M.
Published: March 2003

We point out that the evolution of a quantum system can be considered as a parallel transport of unitary operators in Hilbert spaces along the time with respect to a generalized connection. The different quantum representations of the system are shown to correspond to the choices of cross sections in the principal fiber bundle where the generalized connection is defined. This interpre- tation of time evolution allows us to solve the problem of the formulation of the evolution of a quantum particle in a four-dimensional gauge field.
Solutions of type D possessing a group with null orbits as contractions of the seven-parameter solution
Authors: García D. Alberto , Plebański Jerzy F.
Published: March 2003

It is shown that several type D solutions with null group orbits of local isometries are limiting contractions of the seven-parameter solution.
Mass dependence of Schrödinger wavefunctions for an exponential potential
Authors: Roychoudhury R. K.
Published: March 2003

It is shown that the condition G(r)⩾0 for r⩾0, where G(r) = (∂/∂m)∫r0(u(r))2dr, does not hold in the case of exponential type potentials.
On the effect of space-time isometries on the neutrino field
Authors: Kolassis Charalampos A.
Published: March 2003

We consider a neutrino field in interaction with a space-time admitting an isometry group and we attempt to derive the symmetries imposed on the neutrino flux-vector and on the neutrino field for solutions of the Einstein-Weyl field equations. It is proved that if one of the following two constraints is imposed, (i) the neutrino field is of class E1, (ii) the neutrino flux-vector is collinear with one of the principal null directions of the Weyl tensor, so that Ψ0 = 0, then ℒnξA = −(i/2)sξA and ℒnlμ = 0, where ξA is the neutrino field, lμ is the neutrino flux-vector, nμ is a Killing vector field, and s is a real constant. However, in the cases of a pure-radiation field with diverging rays and a pure-radiation field with nondiverging rays and Ψ3 = 0 the above formulas become ℒnξA = (1/2)(p−is)ξA and ℒnlμ = plμ, where now p and s are in general real functions of the coordinates.
Coulomb-modified nuclear scattering. I
Authors: Talukdar B. , Ghosh D. K. , Sasakawa T.
Published: March 2003

We contemplate deriving a wavefunction approach to Coulomb-distorted nuclear scattering. The theory of ordinary differential equations supplemented by certain well-known properties of higher transcendental functions has been found adequate for the purpose if the nuclear potential is a nonlocal separable one with exponential form factors. The method presented will work for potentials of arbitrary rank. We have derived specific results for Jost function and Fredholm determinants for scattering by (i) Coulomb plus Yamaguchi and (ii) Coulomb plus Mongan case IV potentials.
On the representation of electromagnetic fiels in gyrotropic media in terms of scalar Hertz potentials
Authors: Przeździecki S. , Laprus W.
Published: March 2003

A proof is given of the following theorem: An arbitrary sourceless electromagnetic field determined in a region of a gyrotropic medium can be represented therein in terms of two scalar functions, called the scalar Hertz potentials, that fulfill a system of two second order partial differential equations. Some restrictions are imposed on the region, and their implications are discussed.
Conservation laws for shallow water waves on a sloping beach
Authors: Akyildiz Yilmaz
Published: March 2003

Shallow water waves are governed by a pair of nonlinear partial differential equations. We transfer the associated homogeneous and nonhomogeneous systems (corresponding to constant and sloping depth, respectively) to the hodograph plane, where we find all the nonsimple wave solutions and construct infinitely many polynomial conversation laws. We also establish correspondence between conservation laws and hodograph solutions as well as Bäcklund transformations by using the linear nature of the problems on the hodograph plane.
Molien generating functions, invariants and covariants of magnetic point groups
Authors: Berenson Rhoda
Published: March 2003

The general properties of the Molien generating function and invariant and covariant tensors for the corepresentations of nununitary point groups are presented. The generating functions and (𝒟1,𝒟m) (invariant) and (𝒟r,𝒟m) (covariant) tensors are obtained for the 32 grey point groups and the 58 black-white point groups.
Bäcklund generated solutions of Liouville’s equation and their graphical representations in three spatial dimensions
Authors: Leibbrandt George , Wang Shein-shion , Zamani Nader
Published: March 2003

The Bäcklund–Bianchi method is employed to generate, in three spatial dimensions, the following multiple solutions of Liouville’s equation ∇2α = exp α: The three-wave interaction function α3 and the five-wave interaction function α5. It is verified numerically that α3 satisfies Liouville’s equation to an accuracy of one part in 1014, while α5 satisfies it to one part in 106. The construction of α5 is conditional upon solving ten nonlinear constraint equations. We analyze the complicated structures of α3 and α5 with the help of a three-dimensional plotting routine. It is found that α3 is, surprisingly enough, only characterized by a single ring singularity, while α5 exhibits three ring singularities. It is speculated that the function tanh α3 represents a ring soliton whose shape appears to be preserved in the nonlinear superposition of similar ring solitons. The derivation of Liouville’s solutions α3 and α5 is intimately connected with the auxiliary functions β2 and β4 which solve Laplace’s equation. The latter are also derived and plotted in the paper.
On the projections of representation spaces of the symmetry group on the Minkowski space
Authors: Rzewuski Jan
Published: March 2003

In this paper we generalize the projection of the representation space of the symmetry group SU(2,2)×U(2) on the Minkowski space to arbitrary internal symmetries U(m). The procedure involves certain restrictions on the coordinates of the representation space. Representations of the symmetry group in the restricted space and in the corresponding restricted Hilbert space are constructed.
Splines and the projection collocation method for solving integral equations in scattering theory
Authors: Brannigan M. , Eyre D.
Published: March 2003

This paper investigates the method of projection collocation using cubic B-spline approximants to solve singular integral equations arising in scattering theory. Theoretical error bounds are provided for the approximation which give criteria for estimating the efficiency and convergence of the method. As numerical examples we solve the two-body K-matrix equation with a separable potential and the Reid 1S0 soft-core potential.
Inverse scattering with coinciding-pole reflection coefficients
Authors: Pechenick Kay R. , Cohen Jeffrey M.
Published: March 2003

In inverse scattering theory, algorithms for solving the Gel’fand–Levitan equation normally break down when poles in the reflection coefficient coincide. Here we present a method for treating an arbitrary number of coinciding poles. We give the first explicit solutions for 3, 4, 5, 6, 8, and 10 poles.
Exact solutions of strong gravity in generalized metrics
Authors: Hojman R. , Smailagic A.
Published: March 2003

We consider classical solutions for the strong gravity theory of Salam and Strathdee in a class of metrics with positive, zero, and negative curvature. It turns out that such solutions exist and their relevance for quark confinement is explored. Only metrics with positive curvature (spherical symmetry) give a confining potential in a simple picture of the scalar hadron. This supports the idea of describing the hadron as a closed microuniverse of the strong metric.
Renormalization group hypothesis for critical phenomena theory
Authors: Baker George A.
Published: March 2003

We give various ‘‘nonperturbative’’ results for strong coupling, ultraviolet cut-off removed limits of the bare mass in g0:φ4:d, lattice cut-off, boson field theory. We find that the renormalization-group, unique, strong-coupling, zero-lattice-spacing, double-limit hypothesis has some remarkable consequences, which seem difficult to reconcile with other available information.
Existence and uniqueness of generalized vortices
Authors: Lohe M. A. , van der Hoek John
Published: March 2003

We investigate properties of the static noninteracting vortices determined by equations which generalize the first order Ginzburg–Landau equations. We prove that for each set of n points in the plane a unique solution exists to the first-order equations, with vortex number n. These n points mark the positions of the n vortices and are the only points at which the Higgs field ‖φ‖ vanishes. Regularity properties of the solution are related to those of an arbitrary non-negative function in the theory.
A new mathematical function connected with boundary value problems in kinetic transport theory
Authors: Liu Zhe-ming
Published: March 2003

A new mathematical function connected with solving the Maxwell transport equation by applying the bimodal two-stream relaxing distribution is defined. This new function gives a more correct description of the direct nonequilibrium effect in gas molecular distribution on the macroscopic transferring of moment flux. In this paper, the differential equation satisfied by this function, its recurrence relations, and series or asymptotic expansions in various conditions are formulated. The degrees of approximations for these expansions are discussed.
A refinement of the Thomas–Fermi approximation for the N body problem
Authors: Conlon Joseph G.
Published: March 2003

The author suggests a refinement of the Thomas–Fermi approximation for the ground state energy EN for an N electron atom. It is known that EN can be written asymptotically as N→∞ as EN ∼αN7/3, where α is given by Thomas–Fermi theory. It has been further conjectured that this asymptotic formula may be refined to EN ∼αN7/3+βN6/3+γN5/3. Suggestions for the contributions to βN6/3 and γN5/3 have been made by Dirac and Von Weizsäcker. Here the author uses known results on short-time asymptotics for diffusion equations to obtain a refinement of the Thomas–Fermi approximation which includes the Dirac and Von Weizsäcker corrections. He also obtains new terms. These are related to the scalar curvature of the Jacobi metric corresponding to the Thomas–Fermi potential.
Type-adapted representations of semidirect product groups
Authors: Kasperkovitz P.
Published: March 2003

Irreducible representations of compact groups can be partitioned into three classes (character test +,0,−). This classification is the same for real, complex, and quaternionic representations and in all three cases a peculiar, type-adapted form of the representation matrices may be chosen (t reps). In this paper it is shown how to construct t reps of semidirect products G s g starting with t reps of G and t reps of some covering groups of subgroups of g. The advantage of using t reps shows up in that the factor system of the little cogroups is real in two of three cases and that real, complex, and quaternionic representations are obtained simultaneously. The method is specialized to direct products and generalized to induction from normal subgroups.
Analysis of the effect of surfaces on the tricritical behavior of systems
Authors: Gumbs G.
Published: March 2003

The order parameter φ(z) of the φ6-dominated tricritical free energy functional is calculated for film and half-space geometries. Extrapolation length (Λ) boundary conditions are used to simulate the effect of the surface. Closed-form expressions for φ(z) of a film are given in terms of Weierstrass elliptic functions, or, alternatively, Jacobi elliptic functions. For a half-space, φ(z) is expressed in terms of hyperbolic functions. In the absence of an external field, it is shown that the phase transitions which the system can undergo may be classified as ordinary (Λ>0), urrface (Λ<0), and special (Λ=∞), like the φ4 theory for second-order phase transitions. The critical exponents for the order parameter at the surface are determined for each type of phase transition. A discussion of the free energy for the surface phase is also presented.
Electrostatic structural transitions in a Yukawa-Wigner solid
Authors: Hall George L.
Published: March 2003

A derivation is supplied for a functional relation between the Fuchs energy ε and the Madelung energy S for a Yukawa-Wigner solid (YWS) in which the usual uniform background of a Wigner solid (WS) is replaced by a periodic array of Yukawa charge distributions with variable ‘‘ripple’’ parameter λ allowing the WS and the empty lattice in the limits of small λ and large λ, respectively. It is the zeros of Δε, and not of ΔS, that are relevant for structural transitions between two lattices. It is knwon that 2εWS=SWS, and Medeiros and Mokross incorrectly assumed 2ε=S for the YWS. Here it is first shown by elementary means that the relation between ε and S varies with λ, and then the functional relation is supplied for all λ. When applied to the bcc-fcc system, it is found that Δε has two zeros whereas ΔS has one not equal to either of those of Δε. Starting with small λ, the sequence of lowest energy structures is bcc, fcc, and bcc if these are the only two allowed to compete. The equations for the sc case have not been evaluated, but it is expected that the full sequence for the cubics will be found to be bcc, fcc, and sc, as this author reported for the Gaussian-Wigner solid.
Comments on the static spherically symmetric cosmologies of Ellis, Maartens, and Nel
Authors: Collins C. B.
Published: March 2003

Ellis, Maartens, and Nel have discussed the viability of static spherically symmetric (SSS) cosmologies in general relativity, and in doing so they have studied some of the mathematical aspects of the field equations in that situation. We investigate further these mathematical aspects. Since the field equations correspond to those studied for stellar models, this question is related to previous investigations in that context. In particular, it is shown that conditions at the center of symmetry do not always uniquely determine the space-time geometry; this has relevance to the numerical investigation of stellar systems. Finally, in view of the need to generalize SSS models, some remarks are made on the possibility of relaxing the staticity condition in the case of models that are shear-free.
Null field solutions of the wave equation and certain generalizations
Authors: Collins C. B.
Published: March 2003

The ordinary wave equation in 3+1 dimensions ⧠φ=0, ⧠≡−∂2/∂t2+∂2/∂x2+∂2/∂y2 +∂2/∂z2 admits null field solutions, characterized by ∇φ⋅∇φ=0, ∇φ⋅∇φ≡−(∂φ/∂t)2+(∂φ/∂x)2 +(∂φ/∂y)2+(∂φ/∂z)2 with ∇φ≠0. It is shown that the general null field solution can be obtained from a knowledge of the ‘‘time-transported’’ solutions, i.e., those solutions of the form φ=t−ψ(x,y,z), where ψ satisfies both Laplace’s equation and the eikonal equation in a Euclidean space. We obtain all second-order scalar wave equations of form f(φ,φ;i;i, φ;i; j φ;i; j)=0 (in arbitrary dimension and involving a single potential function φ) for which the above technique applies. These equations are shown to be equivalent to the family of quasilinear third-order equations ∇φ⋅∇(⧠φ)+K(⧠φ)2=0, where K is a constant. Some null solutions of these equations are considered, and related to previous works. The results are applied to determine all shear-free hypersurface-orthogonal null geodesic congruences in Minkowski space–time, and some brief comments are made on complex solutions and on more general wave equations.
Deformation of Lie group representations and linear filters, Part I
Authors: Hermann Robert
Published: March 2003

‘‘Deformation theory’’ is a branch of mathematics which studies the geometry of dependence on parameters of geometric and physical systems. Material arising from Lie group theory and mathematical physics (e.g., in the study of asymptotic behavior of angular momentum) is applied to study the asymptotic behavior of certain linear filters depending on parameters. A mathematical machine which unifies many of these problems will be developed in this series.
The paths of integration for the new generalized Bessel transform
Authors: Bahar E.
Published: March 2003

It is shown that the comments made in a recent paper regarding the previously developed ‘‘New generalized Bessel transform and its relationship to the Fourier, Watson, and Kontorowich-Lebedev transforms’’ are based on erroneous assumptions. The claim that the path of integration parallel to the real axis (below the singularities of the transform function on the real axis) cannot be transformed to a contour around the singularities of the transfer function in the lower half-plane contradicts the very basis of the firmly established Watson transformation and the most advanced theories in radio wave propagation over the Earth’s surface and cylindrical structures.
Thorpe–Hitchin inequality for compact Einstein 4-manifolds of metric signature (++−−) and the generalized Hirzebruch index formula
Authors: Matsushita Yasuo
Published: March 2003

It is proved that the Euler characteristic and the Hirzebruch index of a compact oriented Einstein 4-manifold of metric signature (++−−) satisfy an inequality which is well known as the Thorpe–Hitchin inequality for the case of a Riemannian metric. To derive the inequality, a generalized Hirzebruch formula relating the index to the first pseudo-Pontrjagin number of the manifold is proved. This formula may be contrasted with Chern’s generalized Gauss–Bonnet formula for a pseudo-Riemannian manifold.
On the Hartree–Fock scheme for a pair of adjoint operators
Authors: Froelich Piotr , Löwdin Per-Olov
Published: March 2003

A generalization of the Hartree–Fock scheme for an arbitrary linear operator—and its adjoint—is derived by using the bivariational principle. It is shown that, if the system operator in the transition value is approximated by two Slater determinants, it is determined by a projector ρ, which corresponds to a generalization of the conventional Fock–Dirac density matrix, but which is no longer self-adjoint. The effective one-particle operator then takes the same form as in the conventional theory. The solution of the stability problem for a pair of adjoint effective operators is finally discussed. Numerical applications are performed elsewhere.
When is the Wigner function of multidimensional systems nonnegative?
Authors: Soto Francisco , Claverie Pierre
Published: March 2003

It is shown that, for systems with an arbitrary number of degrees of freedom, a necessary and sufficient condition for the Wigner function to be nonnegative is that the corresponding state wavefunction is the exponential of a quadratic form. This result generalizes the one obtained by Hudson [Rep. Math. Phys. 6, 249 (1974)] for one-dimensional systems.
Aspects of modular quantization
Authors: Kleeman R.
Published: March 2003

A recent method of quantization is examined. A condition is found to isolate suitable Fock representations. Using these representations and a generalized Klein transform the quantization is compared with the normal quantization with U(m) symmetry. Finally the connection of this work with the color superalgebras is shown.
Nonsingular modified Lippmann–Schwinger equation for two charged particles
Authors: Dušek J.
Published: March 2003

We consider the partial waves of the two-body Lippmann–Schwinger equation for the T matrix in the case of the sum of the Coulomb plus a more rapidly vanishing potential. Using the knowledge of explicit forms of the integrals determining the operation of the Lippmann–Schwinger operator on the singularities that characterize the on-shell behavior of the Coulomb-like T matrix, we extract a manifestly nonsingular integral equation from the Lippmann–Schwinger equation. We show that the half- and on-shell inhomogeneous terms and solutions can be made as smooth as the momentum representation partial projection of the non-Coulomb part of the potential.
Theory of light cone cuts of null infinity
Authors: Kozameh Carlos N. , Newman Ezra T.
Published: March 2003

Light-cone cuts of null infinity are defined to be the intersection of the light cone of an interior point xa with the future null boundary of the space-time, i.e., ℐ+. It is shown how from the knowledge of the set of light-cone cuts of ℐ+, the interior (conformal) metric can be reconstructed. Furthermore, a differential equation defined only on ℐ+ is proposed so that (1) the solution space (the parameters defining the set of solutions) is identified with or defines the space-time itself and (2) the solutions themselves yield the light-cone cuts which in turn give metrics conformally equivalent to vacuum solutions of the Einstein equations.
Light cone cuts of null infinity in Schwarzschild geometry
Authors: Joshi P. S. , Kozameh C. N. , Newman E. T.
Published: March 2003

Light cone cuts of future null infinity in Schwarzschild geometry are studied here. The future null cone from an arbitrary apex in the space-time has been constructed, and its intersection with ℐ + is obtained. Knowledge of the cuts yields a great deal of information about the interior of the space-time. In particular, we use it to reconstruct the Schwarzschild metric up to a conformal factor.
All null orbit type D electrovac solutions with cosmological constant
Authors: García D. Alberto , Salazar I. Humberto
Published: March 2003

All null orbit type D solutions of the Einstein–Maxwell equations with λ are obtained. There are only two families of solutions depending upon whether the complex expansion of the electromagnetic eigenvectors aligned along the double DP vectors is different or equal to zero; they are the null orbit solution with complex expansion, and the five-parameter free of complex expansion null orbit solution, respectively.
Character generators for compact semisimple Lie groups
Authors: Gaskell R. W.
Published: March 2003

A new character formula is presented which leads automatically to character generators in the positive form. The character generator for B3 is constructed as an example.
Character generators for elements of finite order in simple Lie groups A1, A2, A3, B2, and G2
Authors: Moody R. V. , Patera J. , Sharp R. T.
Published: March 2003

Elements of finite order in the compact simple Lie groups SU(2), SU(3), SU(4), Sp(4) or O(5), and G(2) are considered. We provide the characters of the elements on irreducible representations of Lie groups by assigning appro- priate numerical values to the variables on which the characters of representations of the Lie group depend. In this way we specialize generating functions for the characters of the representations of the Lie groups to the generating functions for characters of the elements of finite order. Particular attention is paid to rational elements, all of whose characters are integers; they are listed and the generating functions for their characters are obtained in a simplified form from which the characters can be read. Gaussian elements are also studied in detail. Their characters are complex valued with integer real and imaginary parts.
Dirac general covariance and tetrads. I. Clifford and Lie bundles and torsion
Authors: Urani John R. , Barker George P.
Published: March 2003

The isomorphic map of Clifford and Lie bundles to arbitrary coordinate atlases using a global orthonormal tetrad field on a parallelizable space-time is used to construct a fully covariant Dirac spinor theory. The Klein–Gordon equation exhibits a natural spin-torsion coupling of the Einstein–Cartan form, the torsion coming from the tetrad field. The tetrad connection coefficients are explicitly derived in addition to their relationship to the usual Levi–Cività coefficients. Various topological conditions for vanishing torsion are given. The Dirac and adjoint Dirac equations are obtained from a simple Lagrangian and the structure of the adjoint equation is discussed.
Classical theories and nonclassical theories as special cases of a more general theory
Authors: Aerts Dirk
Published: March 2003

We analyze the difference between classical mechanics and quantum mechanics. We come to the conclusion that this difference can be found in the nature of the observables that are considered for the physical system under consideration. Classical mechanics can only describe a certain kind of what we called ‘‘classical observable.’’ Quantum mechanics can only describe another kind of observable; it cannot describe, however, classical observables. To perform this analysis, we use a theory where every kind of observable can be treated and which is in a natural way a generalization of both classical and quantum mechanics. If in a study of a physical system in this theory we restrict ourselves to the classical observables, we rediscover classical mechanics as a kind of first study of the physical system, where all the nonclassical properties are hidden. If we find that this first study is too rough we can also study the nonclassical part of the physical system by a theory which is eventually quantum mechanics.