bluuebunny/crossref_metadata_2025 huggingface dataset:
Null geodesic surfaces and Goldberg–Sachs theorem in complex Riemannian spaces
Authors: Plebañski J. F. , Hacyan S.
Published: March 2003

An extension of the Goldberg–Sachs theorem for the case of a complex V4 is given with a simple proof. The interpretation of the theorem, however, no longer applies the concept of the geodesic and shearless congruence of null directions; instead, the existence of a geodesic 2-surface (complex), the tangent vectorial space to which (i) contains only null vectors, (ii) is parallelly propagated along the surface, is now essential.
Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. IV. Effective−medium theory and cumulant expansion method
Authors: Hori Motoo , Yonezawa Fumiko
Published: March 2003

Several perturbation solutions for the effective permittivity in a completely random medium are evaluated and the validity of the approximations is discussed. It is shown through a diagram technique that the effective−medium theory in a macroscopically inhomogeneous material is equivalent to the coherent−potential approximation in a discordered binary alloy not only in its physical concept but also in its mathematical structure. A cumulant expansion method which substantially takes into account the clustering effects is proposed while the effective−medium theory or the coherent−potential approximation is, by its nature, a single−site approximation and neglects the clustering effects. The numerical results obtained by the effective−medium theory and by the cumulant method for a binary mixture of a conducting material and an insulating material are compared with the computer simulation data on the effective conductivity of a three−dimensional random network. The solution of the cumulant method gives a remarkably good agreement with the computer simulation for the whole range of parameters. An important point is that the cumulant expansion theory holds excellently even near the critical percolation concentration where the clustering plays an essential role and where the effective−medium theory fails to work.
Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. V. One− and two−dimensional systems
Authors: Hori Motoo , Yonezawa Fumiko
Published: March 2003

A perturbation theory is developed for the effective permittivity of one− and two−dimensional random heterogeneous materials that are statistically homogeneous. Closely following the formulations for three−dimensional systems presented in Papers I−IV in this series of work, we derive the formal perturbation solutions for two−dimensional systems, evaluate the second−order and third−order terms for cell materials, and determine upper and lower bounds of the effective permittivity. Here statistical isotropy is not necessarily required. Several approximate perturbation solutions for completely random systems (which are statistically isotropic) are obtained by summing some selected partial series of the perturbation expansion up to an infinite order and numerical results are illustrated. We analyze validity of approximations by means of diagram representation of the perturbation series. The effective−medium theory serves as a good approximation for two−dimensional systems and gives the critical percolation concentration correctly. For one−dimensional materials, the effective−medium approximation turns out to be an exact solution.
The representation of the S
O (4,1) group in four−dimensional Euclidean and spinor space
Authors: Bogdanović R. , Whitehead M. A.
Published: March 2003

An explicit representation of the SO (4,1) group in both SO (4) spinor space and four−dimensional Euclidean space has been found. The two spaces are related by linearly transforming the variables. It has been shown that the four−dimensional hyperspherical harmonics in both four−dimensional polar coordinate systems transform in accordance with the W = 0, Q = 2 representation of the SO (4,1) group. A set of new recursion relations is derived for the SO (3) group reduced rotational matrices, together with a set of standard recursion relations for the Legendre and Gegenbauer polynomials, all of which are obtained by transforming both forms of the SO (4) group basis states in spinor space into four−dimensional space. The matrix elements of the noncompact SO (4,1) group generators are given in the (j,m) basis.
Topology of Higgs fields
Authors: Arafune J. , Freund P. G. O. , Goebel C. J.
Published: March 2003

It is shown that the conserved magnetic charge discovered by ’t Hooft in non−Abelian gauge theories with spontaneous symmetry breaking is not associated with the invariance of the action under a symmetry group. Rather, it is a topological characteristic of an isotriplet of Higgs fields in a three−dimensional space: the Brouwer degree of the mapping between a large sphere in configuration space and the unit sphere in field space provided by the normalized Higgs field ?a = φa (φbφb)−1/2. The use of topological methods in determining magnetic charge configurations is outlined. A peculiar interplay between Dirac strings and zeros of the Higgs field under gauge transformations is pointed out. The monopole−antimonopole system is studied.
The Sagnac effect in general relativity
Authors: Ashtekar Abhay , Magnon Anne
Published: March 2003

The Sagnac effect is a phase shift observed between two beams of light traversing in opposite directions the same closed path around a rotating object. A description of this experiment is obtained within the context of general relativity. In this context the effect provides an operational definition of rotation. An expression for the magnitude of the phase shift is derived under fairly general conditions. The general definition of rotation provided by this experiment is shown to reduce, in certain particular cases, to the usual definitions available. It is observed that the Sagnac effect represents a gravitational analog of the Aharanov−Bohm effect in electrodynamics.
A theorem on the adiabatic scaling of classical orbits
Authors: Witten T. A.
Published: March 2003

The theorem describes the effect on the motion when the Hamiltonian is scaled, that is, when dimensioned parameters in the Hamiltonian are varied without changing any dimensionless parameters in it. The motion changes in a way that is not in general simple or predictable via dimensional analysis. But under certain conditions the motion scales with the Hamiltonian: Classical orbits scale in coordinate space and constants of the motion scale according to their dimensions. The theorem is based on the observation that the observable D which generates dilations may be expressed as a time derivative of a simple quantity. We find it most direct to use quantum mechanics to describe dynamics in the adiabatic limit. In so doing we develop methods that may be useful for other classical adiabatic problems.
The construction of solution of nonlinear relativistic wave equation in λ:Φ44: theory
Authors: Ra̧czka R.
Published: March 2003

The nonlinear equation (⧠+m2) Φ (x) =λ:Φ3: for the quantum scalar field with special form of ordering of the interaction term is considered. The unique soultion Φ (x) of this equation is constructed. It satisfies the relativistic covariance, asymptotic, irredicibility and the relativistic primitive causality conditions in the sense of sesquilinear form. The spectral condition is also satisfied.
Riemann−Green’s functions for solving electromagnetic problems exhibiting rotational symmetry in media moving with superluminal velocities
Authors: Solimeno Salvatore
Published: March 2003

A new method of investigation of the electromagnetic field configuration in media moving with superluminal velocities is illustrated. The basic assumption is the rotational symmetry of the field. It is shown that a Volterra integral equation can be written for the magnetic field on the metallic boundary surface. This equation rests on the knowledge of the Riemann−Green’s functions of two differential operators. Some expressions for these functions are obtained.
Closed gravitational−wave universes: Analytic solutions with two−parameter symmetry
Authors: Gowdy Robert H.
Published: March 2003

Einstein’s vacuum field equations are solved for spacetimes with two−parameter spacelike symmetry, a space−reflection symmetry, and space sections homeomorphic to either S1×S2 or S3. All integrals are evaluated, and the spacetime metrics are presented in analytic form.
On the spinning axis representation
Authors: de Azcárraga J. A. , Navarro L.
Published: March 2003

A new spinning axis representation is introduced. It allows us to calculate the evolution operator of a system with slowly varying time−dependent Hamiltonian with the desired degree of approximation in the parameter used for describing its dynamical evolution. The procedure is compared with a previously existing one and applied to a simple example.
An exact solution of the spin−spin autocorrelation function for a one−dimensional system of hard rods
Authors: Subramanian G. , Levitt D. , Davis H. T.
Published: March 2003

It is shown exactly that, in a one−dimensional system of hard rods with spins, the autocorrelation function of any function of spin F(w) decays as t−1 at long times provided that 〈F〉eq exists and that g (0) ≠ 0, where g (v) is the linear velocity distribution function. As a consequence of this, when F (w) = w, the spin diffusion coefficient defined by the Kubo relation Ds = ℱ∞0 〈w (0) w (t) 〉 dt does not exist. The results are true for arbitrary initial equilibrium velocity and spin distributions, the only restriction being that they be symmetric.
Ray and explosive solutions of nonlinear evolutional equations in Hilbert spaces
Authors: Wang P. K. C.
Published: March 2003

Many nonlinear phenomena occurring in physical systems are describable by a set of ordinary, partial or functional differential equations which can be regarded as an evolutional equation in a suitable abstract vector space. In this paper, we consider nonlinear evolutional equations defined on Hilbert spaces. Attention is focused on developing conditions for the existence of solutions which lie along half−rays emanating from the origin of the space. The results are used to establish sufficient conditions for the existence or nonexistence of explosive solutions or solutions having finite escape time. The paper concludes with a discussion of the application of some of the results to specific classes of evolutional equations arising from physical situations.
Functional integration, Padé approximants, and the Schrödinger equation
Authors: Bove A. , Fano G. , Turchetti G. , Teolis A. G.
Published: March 2003

A method for finding the eigenvalues and the eigenvectors of the Schrödinger Schrödinger equation is presented. If H = T + V is an M−body Hamiltonian, we use Trotter’s formula in the form e−βT/n e−βV/n ∼ e−βH/n (for n large). This allows the computation of the matrix elements of e−βH in the configuration representation, and the moments μr = (ψ,(e−βH)rψ) (r=1,2,⋅⋅⋅) for any wavefunction ψ. From the moments μr we compute the [N − 1/N] Padé approximant, whose poles are the approximate eigenvalues of e−βH. The convergence of the method is proved and asymptotic formulas for the matrix elements of e−βT projected on states of given angular momentum are derived.
Particles and simple scattering theory
Authors: Corbett J. V.
Published: March 2003

A necessary and sufficient condition for a pair of subspaces to be the in and out scattering subspaces for a simple scattering system is obtained. It involves the existence of certain representations of the Galilean presymmetry group on these subspaces. The physical interpretation is that, in scattering, particles remain as particles even in the presence of the interaction.
Degeneracies in energy levels of quantum systems of variable dimensionality
Authors: Herrick D. R.
Published: March 2003

Introduction of variable dimensionality to the Schrödinger equation gives rise to interdimensional degeneracies in the energy levels of one−, two−, and three−electron atoms and molecules. In all cases the degeneracies result from a factorization of the wavefunction into a product of a ’’radial’’ type function times an ’’angular’’ type function. Scaling of the orbital angular momentum quantum number in the one−electron radial equation to obtain ’’excess angular momentum’’ is shown to be equivalent to a variation in dimensionality.
The unitary irreducible representations of SL¯ (3,R)
Authors: Šijački Dj.
Published: March 2003

The unitary irreducible representations of the universal covering group, SL¯ (3,R), of the SL (3,R) group are analyzed by means of the methods developed by Harish−Chandra and Kihlberg. We have found a single closed expression for the matrix elements of the noncompact generators for an arbitrary unitary representation of the SL¯ (3,R) group. The irreducibility of the representations is achieved by using the little group technique and the scalar product for each irreducible Hilbert space is explicitly given. Contraction (in the Inönü and Wigner sense) of the SL¯ (3,R) unitary irreducible representations to the corresponding representations of the T5 σ SU (2) group is discussed.
Representation theory of the universal covering of the Euclidean conformal group and conformal invariant Green’s functions
Authors: Grensing G.
Published: March 2003

We present a special realization for the universal covering of the Euclidean conformal group. This group can be defined as a transformation group on the Euclidean version of compactified Minkowski space such that the action on this space coincides with the usual one of the Euclidean conformal group. We construct the representations of the principal and complementary series and derive the intertwining kernels for the equivalent representations. The connection between representation theory and conformal invariant quantum field theory is studied. To this end we also give the reduction of the tensor product of two representations of the supplementary series.
Novel identities for simple n−j symbols
Authors: Dunlap B. I. , Judd B. R.
Published: March 2003

A number of identities involving 3−j, 6−j, and 9−j symbols in combinations not commonly encountered in angular−momentum theory are established and briefly discussed.
Spacetime symmetries and the complexion of the electromagnetic field
Authors: Ray John R. , Thompson Everette L.
Published: March 2003

Considering spacetimes admitting groups of motions, we derive the symmetry imposed on the electromagnetic field for solutions to the Einstein−Maxwell equations. It is found that the scalar products between the gradient of the complexion and the Killing vectors must be constants and that these constants enter into the symmetry condition on the electromagnetic field. We therefore obtain a geometrical interpretation of the complexion in spacetimes having symmetries.
Algebraic identities among U (n) infinitesimal generators
Authors: Okubo Susumu
Published: March 2003

Some algebraic identities among infinitesimal generators of the n−dimensional unitary group U (n) have been found. They satisfy a simple quadratic equation for degenerate representations. A generalization of Holstein−Primakoff boson realization for the U (n) group is also given.
Some remarks on the evolution of a Schrödinger particle in an attractive 1/r2 potential
Authors: Radin Charles
Published: March 2003

Comparing the different solutions of Case and Nelson for the evolution operators of a Schrödinger particle in the potential V (r) = −1/r2, we show that Nelson’s nonunitary solution is a simple average, over a physical parameter related to a boundary condition at the singularity, of Case’s family of solutions.
Time−dependent dynamical symmetries, associated constants of motion, and symmetry deformations of the Hamiltonian in classical particle systems
Authors: Katzin Gerald H. , Levine Jack
Published: March 2003

A study is made of time−dependent dynamical symmetry mappings of Hamilton’s equations for classical particle systems. The conditions that an infinitesimal mapping (δxA, δt) ≡ (ξA(x,t) δa, ξ0(x,t) δa), A = 1, ..., 2n, be a symmetry mapping are expressed in terms of a ’’symmetry vector’’ ZA(x,t) ≡ ξA − ξ0 ηAB∂BH (x,t) (where ηAB defines the symplectic matrix of phase space). These conditions imply that ξ0 is arbitrary. It is shown that the symmetry deformation of a constant of motion M (x,t) will also be a (’’derived’’) constant of motion (time−dependent related integral theorem). It follows for the case H = H (x) that every time−dependent symmetry deformation of H (x) is a constant of motion, and it is shown conversely that every constant of motion M (x,t) can be expressed as a symmetry deformation of the Hamiltonian, that is, there exists a symmetry vector ZA(x,t) such that M = ZA∂AH. It is found that if ZA(≠0) is a symmetry vector, then M (x,t) ZA will be a (scaled) symmetry vector if and only if M is a constant of motion. The existence of groups of symmetry vectors is considered, and it is shown that a complete set of r symmetry vectors ZAα, α = 1, ..., r, determines an r−parameter continuous group of symmetries. A special class of symmetry vectors ZA(P) (x,t) ≡ ηAB (∂BM − N∂BH), (’’extended Poisson vectors’’), where M (x,t), N (x,t) are constants of motion is defined and conditions that such vectors determine a symmetry group are obtained. Poisson vectors are also used to show that the related integral theorem mentioned above may be considered as a generalization of Poisson’s theorem on constants of motion. Dependency relations between derived constants of motion with respect to vectors of a symmetry group are obtained.
Observables, operators, and complex numbers in the Dirac theory
Authors: Hestenes David
Published: March 2003

The geometrical formulation of the Dirac theory with spacetime algebra is shown to be equivalent to the usual matrix formalism. Imaginary numbers in the Dirac theory are shown to be related to the spin tensor. The relation of observables to operators and the wavefunction is analyzed in detail and compared with some purportedly general principles of quantum mechanics. An exact formulation of Larmor and Thomas precessions in the Dirac theory is given for the first time. Finally, some basic relations among local observables in the nonrelativistic limit are determined.
Consistency in the formulation of the Dirac, Pauli, and Schrödinger theories
Authors: Gurtler R. , Hestenes D.
Published: March 2003

Properties of observables in the Pauli and Schrödinger theories and first order relativistic approximations to them are derived from the Dirac theory. They are found to be inconsistent with customary interpretations in many respects. For example, failure to identify the ’’Darwin term’’ as the s−state spin−orbit energy in conventional treatments of the hydrogen atom is traced to a failure to distinguish between charge and momentum flow in the theory. Consistency with the Dirac theory is shown to imply that the Schrödinger equation describes not a spinless particle as universally assumed, but a particle in a spin eigenstate. The bearing of spin on the interpretation of the Schrödinger theory is discussed. Conservation laws of the Dirac theory are formulated in terms of relative variables, and used to derive virial theorems and the corresponding conservation laws in the Pauli−Schrödinger theory.
On first order smoothing theory
Authors: Roberts P. H. , Soward A. M.
Published: March 2003

This paper is concerned with the validity of an approximation known variously as first order smoothing theory (FOST), first order cumulant discard, quasilinear theory, and the adiabatic approximation. This approximation has been widely used in turbulent dynamo theory, which provides a number of the examples considered below. On the basis of such examples, Lerche and Parker have recently contended that first order smoothing is under no circumstances trustworthy; i.e., even when the method might have been expected to yield solutions in agreement with those of the exact theory, it does not necessarily do so. We show that the discrepancies between the exact and approximate theories that they claim to have uncovered are, in fact, nonexistent. It is established, however, from one of their examples that FOST may be legitimately criticized on a completely different ground connected with the high velocity tail of the probability distribution.
Lattice sums and scattering coefficients for the rectangular planar array
Authors: Twersky Victor
Published: March 2003

We consider the plane wave mode solution for the three−dimensional problem of multiple scattering by a doubly periodic planar array of bounded obstacles. The mode amplitudes are expressed in terms of the multiple scattering coefficients for one obstacle of the array, and these are specified algebraically by the single scattered coefficients and the spherical wave lattice sums that characterize the array. We derive rapidly convergent forms of the lattice sums by exploiting our earlier results for the periodic line of bounded obstacles and for the grating of parallel cylinders. Then we develop closed form approximations for the multiple scattering coefficients for small scatterers which exhibit the effects of the array in coupling the multipole coefficients of the isolated scatterers.
Perturbative−variational approximations to the spectral properties of semibounded Hilbert space operators, based on the moment problem with finite or diverging moments. Application to quantum mechanical systems
Authors: Bessis D. , Villani M.
Published: March 2003

We introduce in a systematic way the properly defined arctangent of the mean value of the resolvent of a Hilbert space operator. We consider more precisely the case of semibounded self−adjoint operators H, in the region of the discrete spectrum. The arctangent of the Padé approximations to the mean value of the resolvent are constructed out of the moments. They are shown to provide converging monotonic increasing sequences of lower bounds to the arctangent of the resolvent. Consequently bounds with the same properties are derived for the discrete eigenvalues of H and, most remarkable, the ordering of the corresponding poles of the approximations reproduces the ordering of the exact poles. The Padé method is shown to provide a way to fully exploit the content of the Rayleigh−Ritz variational method, by providing a simple mechanical procedure to build up the variational subspaces: It defines a powerful both perturbative and variational approximation to semibounded operators. The difficulties of the Ritz method in the degenerate case are overcome by the fact that all bounds in the Padé method are strict bounds. In a second part, we consider the important case in which the moments are given by diverging algorithms. By properly regularizing them, we show that the Padé−Ritz variational principle generalizes to produce absolute maxima of the arctangent of the Padé approximations in the regulator and that these maxima form monotonic converging sequences of lower bounds. In the last part, we discuss an application to quantum mechanical systems for which the perturbative variational method is applied to the energy, allowing us to treat the case of strong coupling. As a consequence it appears possible to solve (approximately) the R−dimensional anharmonic oscillator in a purely algebraic way.
Static cylindrically symmetric solutions of the Einstein−Maxwell equations
Authors: Chitre D. M. , Güven R. , Nutku Y.
Published: March 2003

A general class of solutions of Einstein−Maxwell equations with static cylindrical symmetry is obtained. The equations are derived using canonical methods and the fields are shown to satisfy a certain Painlevé differential equation of the third transcendental type. A particular algebraic solution is studied in detail, and is found to have a certain mass and current on the axis and a helical magnetic field around it.
Boundaries of spacetimes
Authors: Duncan D. P. , Shepley L. C.
Published: March 2003

A review of the Schmidt technique and the Sachs technique for assigning boundary points to spacetimes is given. A modification of the Sachs process which makes it obviously identical with Schmidt’s is suggested. Some simple examples are discussed.
Phase shifts as functions of the cross section
Authors: Atkinson D. , Kaekebeke M. , de Roo M.
Published: March 2003

We show that an elastic scattering amplitute may be defined as an implicit function of the differential cross section. A practical method is given for a numerical investigation of this dependence, both in the elastic and in the inelastic regions. In particular, we consider the case of a polynomial amplitude, and we show that the Crichton ambiguity is not isolated.
Conserved vectors of the Komar type and compatibility identities in Lagrangian field theories
Authors: Pavelle Richard
Published: March 2003

Given a Lagrangian which is an invariant density and a concomitant of a set of independent geometrical objects, it is shown that there is a unique conserved vector associated with the Lagrangian. For cases of interest the vector is constructed and is seen to be identical with Komar’s conserved vector for general relativity. For a general Lagrangian we also construct the ’’compatability identity’’ relating the Euler−Lagrange tensors. The connection between these identities, the invariance identities of Rund, and Lie derivatives of invariant densities is found.
Path−integral evaluation of a nonstationary Calogero model
Authors: Goovaerts M. J.
Published: March 2003

The path integral for a three−body problem in one dimension described by the Lagrangian L = (1/2) {?12 + ?22 + ?32} − (1/4) ω2(t) { (ξ1 − ξ2)2 + (ξ2 − ξ3)2 + (ξ3 − ξ1)2} − g (ξ1 − ξ2)−2 is examined, where ξ1, ξ2, and ξ3 are the coordinates of the three particles and ω (t) is the time−dependent angular frequency indicating the coupling strength of the harmonic potentials.
Systems of differential inequalities and stochastic differential equations. II
Authors: Ladde G. S.
Published: March 2003

By employing vector Lyapunov−like functions and the theory of systems of differential inequalities, a very general comparison theorem for Itô type stochastic differential systems is developed. Furthermore, sufficient conditions are given for conditional stability and conditional boundedness of solutions in the mean.
Space−time symmetries and nonunitary representations of the Poincaré semigroup: Description of unstable particles
Authors: Comi M. , Lanz L. , Lugiato L. A. , Ramella G.
Published: March 2003

In this paper we justify the use of nonunitary representations of the space−time symmetry group to describe physical systems, as, e.g., unstable particles. It is shown that such a generalization of the standard theory of symmetries is possible inside the approach to quantum mechanics recently developed by Ludwig. The Poincaré semigroup turns out to be the fundamental symmetry group and only Lorentz transformations must be unitary.
Correction terms for Padé approximants
Authors: Barnsley M. F.
Published: March 2003

Let f(x) be a function of the form f(x) = ∫∞0dφ (u)/(1+ux), x⩾0, where φ (u) is of bounded variation and piecewise differentiable on 0⩽u<∞; and suppose that the [N−1/N] and [N/N] Padé approximants (PA’s) to f(x) can be constructed. Then correction terms bN(x) and cN(x) such that [N−1/N]−bN(x) ⩽ f(x) ⩽ [N/N]+cN(x), x⩾0, are considered. Suitable corrections are shown to have the structure x2N{positive constant}/{denominator of corresponding PA}2. The nature of the constants is examined: They vanish when f(x) is a series of Stieltjes, and given appropriate information about f(x) they can be calculated. Applications are considered.
Exact solutions to Einstein field equations
Authors: Melnick Jorge , Tabensky Romualdo
Published: March 2003

A method of generating static and isotropic solutions of the Einstein field equations is presented for the case in which the source of the field is a perfect fluid. A special case is exhibited. The general solution for vacuum is derived.
Scattering by singular potentials with a perturbation—Theoretical introduction to Mathieu functions
Authors: Aly H. H. , Müller−Kirsten H. J. W. , Vahedi−Faridi N.
Published: March 2003

Different treatments of the important potential 1/r4 are correlated to understand their interrelations and to clear up the connection between the eigenvalues of Mathieu’s equation and the poles of the S matrix. We also derive a new solution of the modified Mathieu equation. Mathematical and physical implications are also discussed.
Quantum mechanics on homogeneous spaces
Authors: Doebner H. D. , Tolar J.
Published: March 2003

A complete description of quantum kinematics on a homogeneous G−space M is presented using imprimitivity systems for G based on M. The kinematics on M is considered (if possible and consistent with this quantization) as kinematics on a G−orbit equivalent to M in some Euclidean space Rn. This method gives a physically justified and mathematically well−defined method of connecting the free Hamiltonian of a quantum system in Rn with an operator proportional to the Laplace−Beltrami operator on M (with the Riemannian structure inherited from Rn) which is proposed to be the free Hamiltonian on M.
On the duality condition for a Hermitian scalar field
Authors: Bisognano Joseph J. , Wichmann Eyvind H.
Published: March 2003

A general Hermitian scalar field, assumed to be an operator−valued tempered distribution, is considered. A theorem which relates certain complex Lorentz transformations to the TCP transformation is stated and proved. With reference to this theorem, duality conditions are considered, and it is shown that such conditions hold under various physically reasonable assumptions about the field. A theorem analogous to Borchers’ theorem on relatively local fields is stated and proved. Local internal symmetries are discussed, and it is shown that any such symmetry commutes with the Poincaré
Matrix elements of the generators of I
U (n) and I
O (n) and respective deformations to U (n,1) and O (n,1)
Authors: Wong M. K. F. , Yeh Hsin−Yang
Published: March 2003

The unitary continuous representations of U (n,1) and O (n,1) are discussed from the point of view of deformation of IU (n) and IO (n). It is shown that there are two general ways of writing the matrix elements of the infinitesimal generators of the groups U (n,1) and O (n,1). The first one is to write them as either pure real or pure imaginary. The second one is to write them as complex. We show how these different ways are related to each other.
Convergence of Padé approximants using the solution of linear functional equations
Authors: Baker George A.
Published: March 2003

We prove that a projection of the solutions to a linear functional equation of the Fredholm type with a compact kernel, projected into the Cini−Fubini subspaces, converge strongly to the solution in the whole space. Here either the whole sequence converges for all nonsingular points of the functional equation with at most one exceptional point, or by selecting at most two infinite subsequences we can obtain convergence for all nonsingular points. We then prove that the diagonal Padé approximants to the inner product of the solution with another element converge. For certain kernels of trace class, the numerator and denominator separately converge. As applications of these results, we prove the pointwise convergence of the Padé approximants to a wide class of meromorphic functions. We also prove the convergence, for decent potentials, of the Padé approximants to the scattering amplitudes for nonrelativistic quantum mechanical scattering problems. The numerators and denominators of the Padé approximants to the partial wave scattering amplitudes for single signed potentials converge separately to entire
Experimental uncertainties in the problem of the unitarity equation
Authors: Tortorella Michael
Published: March 2003

In this article the question of how experimental uncertainties affect the construction of the scattering amplitude from the differential cross section and unitarity at a fixed energy is examined. It is shown that in most cases in which the solution can be found by the method of Newton and Martin, the problem is ’’well−posed’’ in the sense that the solution depends continuously on the data. A new proof is given of the fact that if the differential cross section is nearly constant and small enough, there is a unique solution of the problem. Some estimates for the scattering amplitude
Algebraic approach of the infrared−problem for external currents
Authors: Provost J. P. , Rocca F. , Vallée G.
Published: March 2003

The infrared problem for external currents is shown to be a consequence of the nonexistence of a particle number in the correct representation. A
NUT−like generalization of axisymmetric gravitational fields
Authors: Reina C. , Treves A.
Published: March 2003

The complex potential formulation of the axisymmetric problem discussed by Ernst enables us to construct new solutions from a given one, by multiplying the corresponding potential by a unit complex number. This rotation introduces naturally the NUT parameter in the metric. The generalized Kerr, Weyl, and Tomimatsu−Sato solutions are explicity constructed.
Nonexistence of dissipative structure solutions to Volterra many−species models
Authors: Rosen Gerald
Published: March 2003

Subject to boundary conditions of practical interest, the only temporally periodic solutions that may be admitted by a generic system of Volterra n−species reaction−diffusion equations are spatially uniform solutions, and thus dissipative structures are precluded as solutions to Volterra n−species models.
N−variable rational approximants and method of moments
Authors: Alabiso C. , Butera P.
Published: March 2003

The method of moments is applied to pairs of linear permutable self−adjoint operators A and B in a Hilbert space ℋ. An approximate expression for the diagonal matrix elements of the operator (1 − wA − zB)−1, where w, z are complex numbers, is taken as a guide to the definition of rational approximants from general formal power series in two variables. Starting from an operator convergence theorem in a certain Hilbert space, we prove the convergence of our approximants to analytic functions of two complex variables with the integral representation G (w,z) = ∫∫ dσ (α,β) / (1 −wα − zβ), under suitable restrictions on the positive measure σ (α,β). The same approximation scheme can also be applied to the diogonal matrix elements of the operator [(1 − wA) (1 − zB)]−1, leading to a different rational approximant which we prove to converge to functions with the integral representation G̃ (w,z) = ∫∫ dσ (α,β) / (1 − wα) (1 − zβ). In both cases the convergence is uniform on appropriate compact subsets of C2. The extension to the n−dimensional case is straightforward for both approximants. The connections with a standard variational principle are also briefly discussed.
Nonspreading solutions of the inhomogeneous scalar wave equation
Authors: Couch W. E. , Torrence R. J.
Published: March 2003

A simple condition that is necessary and sufficient for the solution of the inhomogeneous wave equation to be a nonspreading wave is derived for a class of driving terms that arise in certain physical problems. The condition is applied to the analysis of the self−scattering of gravitational multipole radiation at second perturbative order. It is proved that there is no scattering at the multipole component of highest order in the second−order gravitational field. It is conjectured that there is no scattering for every component of the second−order field. A mathematical expression of this conjecture, derived from the condition for nonspreading, is given and it implies conjectured identities on Clebsch−Gordan coefficients.
Theory of Regge poles for 1/r2 potentials. I
Authors: Mastalir Robert O.
Published: March 2003

In the following series of papers we give a detailed study of the theory of Regge poles for 1/r2 potentials with the behavior r2 V (r) = −V0 at r = 0 and r2 V (r) = −V2 at r = ∞. We give a complete description of the distribution of the Regge poles in the λ plane, which is cut from −V01/2 to V01/2 and study the behavior of the pole trajectories. We find that the high energy limit of the Regge poles is controlled by the parameter p = (λ2−V0)1/2 while at low energies the relevant parameter is q = (λ2−V2)1/2. This means that the point λ = 0 for the case of Yukawa potentials corresponds here to the point q = 0. We also find that the Regge trajectories λ (E) may have branch points of the square root type at finite, in general complex, values of E at which points the pole passes the origin λ = 0. We further find that the kinematic singularity of the S matrix at k = 0 is more complicated than it is for Yukawa potentials and is here characterized by the Floquet parameter ν (λ,k) associated with the Schrödinger equation. We illustrate these and other results with some
Double coset analysis for symmetry adapting Nth rank tensors of U (n) to its unitary subgroups
Authors: Sullivan John J.
Published: March 2003

Representations of the unitary group U (n) symmetry adapted to the subgroup sequences U (n) U(nj)U(in) U (inj) are considered using double coset decomposition. The matrix elements of the double coset representatives are related to identical coefficients developed
Temporally inhomogeneous scattering theory for modified wave operators
Authors: Hendrickson Joseph H.
Published: March 2003

A theorem of Alsholm and Kato, which gives existence of modified wave operators for a large class of long range potentials, is extended to include time dependent potentials. It is then shown that these temporally inhomogeneous modified wave operators WD±(S) vary continuously, in some sense, on the potentials. This result is new for both time dependent and time independent potentials. In addition, part of the nonuniqueness problem of modified wave operators is resolved by noting that the modified wave operators
Models of Hamiltonian several-particle interactions
Authors: Arens Richard
Published: March 2003

A method of Hamiltonian surfaces proposed recently as a semiclassical completely Hamiltonian way of defining the motions of N particles (with finitely many degress of freedom) is shown to be mathematically feasible for N ⩾ 2. The interactions are invariant under the Poincaré group. The possibility of describing each motion by N world lines, the same for all observers, is dispensed with. The usual commutation relations are retained.
Bifurcate nondiverging null hypersurfaces and trapped surfaces
Authors: Demmie Paul N.
Published: March 2003

An empty spacetime containing a bifurcate nondiverging null hypersurface is investigated, and conditions are given that are necessary and sufficient for the existence of trapped surfaces near this hypersurface to its future. These conditions involve a topological requirement that the two-surface of bifurcation be compact and an inequality that must be satisfied by the characteristic data for this spacetime—the metric of the two-surface of bifurcation and an arbitrary function given on this surface that is shown to be related to angular momentum. The existence of a bifurcate Killing horizon in this spacetime is established. Finally, a Kerr spacetime containing bifurcate Killing horizons is examined, and results pertaining to the existence of trapped surfaces near these horizons to their futures are obtained. These results involve the parameters representing mass and angular momentum per unit mass.
Variational principles, variational identities, and supervariational principles for wavefunctions
Authors: Gerjuoy E. , Rau A. R. P. , Rosenberg L. , Spruch Larry
Published: March 2003

We develop variational principles and variational identities for bound state and continuum wavefunctions in a general context, paying particular attention to the proper choice of defining equations and boundary conditions which will lead to unique and unambiguous wavefunctions even when these functions are complex. Any functional, such as a matrix element, calculated with such a variationally determined wavefunction, will also be accurate to second order in the error of the starting choice. This provides, therefore, an alternative procedure for getting variational estimates of matrix elements to the one that already exists in the literature and we establish the equivalence of the two. Of even more interest is the possibility which now seems open of going beyond the variational principle and generating ’’supervariational’’ estimates of wavefunctions and matrix elements which are good to better than second order. We also give a simple prescription for the construction of variational identities for wavefunctions, that is, identities which lead readily to variational principles and, more significantly, might well serve as a starting point for the development of variational bounds.
Global existence of solutions to the Cauchy problem for time-dependent Hartree equations
Authors: Chadam J. M. , Glassey R. T.
Published: March 2003

The existence of global solutions to the Cauchy problem for time-dependent Hartree equations for N electrons is established. The solution is shown to have a uniformly bounded H1(R3) norm and to satisfy an estimate of the form ∥ ψ (t) ∥H2 ⩽ c exp(kt). It is shown that ’’negative energy’’ solutions do not converge uniformly to zero as t → ∞.
The classical nonlinear oscillator and the coherent state
Authors: Bhaumik K. , Dutta-Roy B.
Published: March 2003

The coherent state constructed out of quantum oscillator states is employed to develop a method for solving the classical nonlinear oscillator problem. The perturbation solution of the Duffing oscillator is used to illustrate the method and to obtain the result of the classical procedure.
O (4,2) symmetry and the classical Kepler problem
Authors: Tripathy K. C. , Gupta Rakesh , Anand J. D.
Published: March 2003

The spectrum generating algebra associated with the negative energy motions of a classical dynamical system, namely the Kepler problem, has been systematically studied with the aid of the Poisson brackets. The canonical map between our realization and that of an earlier analysis by Barut and Bornzin has been established.
On the relaxation to quantum-statistical equilibrium of the Wigner–Weisskopf atom in a one-dimensional radiation field. VII. Emission in a finite system in the presence of an extra photon
Authors: Davidson Russell , Kozak John J.
Published: March 2003

In this paper, we present the exact solution to a problem previously unsolved in radiation theory: the emission of a two-level atom in a (one-dimensional) radiation field in the presence of an extra photon. The solution is obtained directly from the Schrödinger equation of the problem using techniques suggested by the work of Muskhelishvili on singular integral equations. The solution corresponding to a finite system as well as the one corresponding to a system infinite in extent are given, although our primary concern in this paper is the finite-system problem. For a particular choice of initial condition, the probability at time t that the two-level atom is in the excited state is found, and the effects of system size and choice of coupling function are studied numerically for a given coupling constant. Our results are compared with those obtained in an earlier paper of the series, wherein the spontaneous emission of a Wigner-Weisskopf atom in a (one-dimensional) field of radiation for a system finite in extent was studied. The physical effects calculated and the conclusions drawn from our comparative studies are all in accord with simple intuition regarding the problem. The paper concludes with some brief remarks on problems in radiation theory which are accessible to study given the methods laid down in this paper.
An integral equation for the Gel’fand–Levitan kernel in terms of the scattering potential in the one-dimensional case
Authors: Moses H. E.
Published: March 2003

An integral equation for the Gel’fand–Levitan kernel is given in terms of the scattering potential. This integral equation may be regarded as complementary to the Gel’fand–Levitan equation which is an integral equation for the kernel in terms of the Fourier transform of the reflection coefficient.
Variable dimensionality in the group-theoretic prediction of configuration mixings for doubly-excited helium
Authors: Herrick David R.
Published: March 2003

Variable dimensionality (= D) is used to interpret recent group-theoretic predictions of configuration mixings in doubly-excited helium. Calculated 2s2:2p2 1S mixings agree with the group theory over the range 1 ⩽ D ⩽ ∞. General results for D = 2 predicted mixings are given and energies of states within the N = 2 atomic shell confirm predicted level orderings. The D = 1 model atom is described exactly by the group theory, with quantum numbers and a selection rule characterizing the stability of Coulomb matrix elements as D → 1. The exact D = 1 results have a physical interpretation in approximate autoionization and energy selection rules for Rydberg series at D = 3.
On the transformation from random Schrödinger Hamiltonians to random Hamiltonian matrices
Authors: McDonald J. F. , Favro L. D.
Published: March 2003

In this article we consider the effects on ensembles of random Hamiltonian matrices when certain restrictions are imposed on the potentials of the corresponding Schrödinger Hamiltonians. In particular, we investigate the validity of the assumption which is usually made when ensembles of random matrices are used to predict statistical properties of energy levels of complex systems, namely that the Hamiltonian matrix elements are statistically independent.
Two-level radiative systems and perturbation theory
Authors: Hioe F. T. , Montroll E. W.
Published: March 2003

We obtain expressions for the radiative level shifts of a two-level system in terms of (i) recurrence relations, (ii) ratios of determinants, (iii) continued fractions, and (iv) a Lidstone expansion. These expressions are shown to be very useful for numerical computations. It is pointed out that perturbation series in powers of the coupling constant are not the most appropriate way of representing the solutions of the problem, but if they are to be used, different series should be employed depending on the relative value of the frequency of the field to the frequency of the two-level system. The significance of these results in the general theory of perturbation is discussed.
The general relativistic fields of a charged rotating source
Authors: Finkelstein Robert J.
Published: March 2003

The gravitational and electromagnetic fields of a charged, rotating source are obtained by an elementary algebraic method.
The notion of essential locality for nonlocalizable fields
Authors: Bümmerstede J. , Lücke W.
Published: March 2003

The classification of extension of the field commutator outside the light cone suggested by Constantinescu and Taylor is analyzed and shown to be to a large extent mathematically equivalent to the notion of essential locality, introduced in a recent paper by the present authors. Simple model fields are constructed which disprove the interpretation given by Constantinescu and Taylor. Essential locality is shown to hold for the two-point function of every scalar Hermitian field, including the massless case. It is, moreover, shown to be weaker than locality and independent of the other Wightman axioms. Unfortunately, essential locality turns out to be unstable under limits. In order to indicate the possibility that there are essentially local fields which do not fall into Jaffe’s class and the commutator of which is concentrated in the closed light cone, Jaffe’s concept of strict localizability is generalized. As a by-product it is indicated that local fields (in the generalized sense) may have extreme high energy behavior.
The separating topology for the Lorentz group L
Authors: Vroegindeweij P. G.
Published: March 2003

Some properties of the Lorentz group L are presented if it is endowed with a topology induced by one of the topologies for the Minkowski space M, proposed by E. C. Zeeman.