bluuebunny/crossref_metadata_2025 huggingface dataset:
Numerical computation of off-shell two-body Coulomb amplitude values
Authors: Sramek Stanley J.
Published: March 2003

Practical methods of computing numerical values for off-shell two-body Coulomb scattering amplitudes are developed. The methods are based on well-known analytic representations of the amplitudes. The results of a comparative computational study of the different methods are presented. It is found that the methods produce numerical values sufficiently precise for use in treating many-body problems formulated in terms of two-body amplitudes.
The planar approximation. II
Authors: Itzykson C. , Zuber J.-B.
Published: March 2003

The planar approximation is reconsidered. It is shown that a saddle point method is ineffective, due to the large number of degrees of freedom. The problem of eliminating angular variables is illustrated on a simple model coupling two N×N matrices.
Projective covering group versus representation groups
Authors: Cariñena José F. , Santander Mariano
Published: March 2003

The relation between two methods of finding the unitary projective representations of a connected Lie group is studied. It is shown that the one of the projective covering group G̃ is singled out as the simplest one, but one could also get a connected and simply connected Lie splitting group of minimal dimension by using a similar procedure.
Path integral in the representation of SU(2) coherent state and classical dynamics in a generalized phase space
Authors: Kuratsuji Hiroshi , Suzuki Tōru
Published: March 2003

Path integral in the representation of coherent state for the simplest semisimple Lie group SU(2) and its classical consequences are investigated. Using the completeness relation of the coherent state, we derive a path integral expression for the transition amplitude which connects a pair of SU(2) coherent states. In the classical limit we arrive at a canonical equation of motion in a ’’curved phase space’’ (two-dimensional sphere) which reproduces the ordinary Euler’s equation of a rigid body when applied to a rotator.
Solution of boundary value problems with Laplace’s equation for ellipsoids and elliptical cylinders
Authors: Eyges Leonard
Published: March 2003

If a dielectric ellipsoid or elliptical cylinder is placed in a uniform applied field it is well known that the field inside remains uniform, and is changed only by a depolarization factor that multiplies each applied field component. This paper generalizes this result. Namely, if for the three-dimensional case the potential Φapp of the applied field can be expanded in the neighborhood of the ellipsoid as Φapp=𝒥l=0L𝒥m=−llDlm rl Ylm(ϑ,φ) where l goes from zero to a maximum value L, then it is shown that the resultant potential inside the ellipsoid, Φint, is Φint=𝒥l=0L𝒥m=−ll ClmrlYlm(ϑ,φ) where the coefficients Clm are found explicitly and there is no Clm with l≳L. For a dielectric constant ε, the limits of the above solution as ε→∞ and ε→o are considered and are shown to yield respectively the solutions to the Dirichlet problem with potential zero on the boundary (grounded perfect conductor) and the Neumann problem with normal derivative of the potential zero on the boundary (ideal fluid flow). The homogeneous problem of free charge on an ellipsoidal perfect conductor is considered and it is shown to require a modification of the methods that yielded the results above. The modified method is applied to the problem of the oblate ellipsoid, and its limiting case of a disc, and it enables the easy derivation of various classical results due to Copson and others on the ’’problem of the electrified disc.’’ Finally, multiple dielectric ellipsoids or elliptic cylinders are considered and it is shown that problems involving such bodies can be solved in powers of ai/dij where ai is a typical length of the ith body and dij is the distance between the ith and the jth. This opens the way, in the proper limit of ε, to the solution of a variety of problems, such as flow around multiple strips and through the slots they may form, penetration of the electric field through perforated screens, and so on.
Dynamic polarizability of the double-quadratic kink
Authors: Trullinger S. E.
Published: March 2003

We discuss a simple model of ferroelectric domain walls in terms of solitary-wave (kink) solutions of the equation of motion for the double-quadratic (DQ) chain. The response of the DQ kink to an external field is calculated and a very simple expression is obtained for the dynamic polarizability of the DQ kink. The simplicity of this model makes it more attractive than the usual φ4 model for studying the interaction of ferroelectric domain walls with external electric fields.
Weakly nonlinear theory of coherent Langmuir waves. I. Wave wave interaction
Authors: Trocheris M.
Published: March 2003

A weakly nonlinear analysis of the Vlasov equation is made in the case of small amplitude Langmuir waves. The nonlinear terms are treated as a small perturbation in the framework of the asymptotic theory of Krylov and Bogoliubov. In order to apply this method, the Vlasov system of equations is transformed by taking as new unknown functions a complete system of constants of the motions of the linearized equations. Such a system can be found with the help of essentially Van Kampen’s and Case’s expansion in normal modes. Resonant wave particle interaction is avoided by cutting off the distribution functions in velocity space and by considering waves of phase velocities larger than the cutoff. In this case, the main physical effect is mode coupling and Davidson’s nonlinear system of equations on the wave amplitudes is recovered. The main point is that the derivation is made without neglecting the free streaming portions of the distribution functions. A discussion of the validity of the approximation and of the relevant time scales is presented.
Weakly nonlinear theory of coherent Langmuir waves. II. Wave particle interaction
Authors: Trocheris M.
Published: March 2003

A weakly nonlinear analysis was carried out in Part 1 in the case of Langmuir waves, by neglecting the tail of the distribution function where particle velocities can be equal to phase velocities of waves. The treatment is extended in Part 2 by including particles of all velocities, so that resonant wave particle interaction becomes possible. The asymptotic theory of Krylov and Bogolioubov is applied to the Vlasov system of equations after writing it in terms of unknown functions which are constants of the motion of the linearized equations. A system of such constants can be found with the help of a suitable expansion in normal modes, which differs from Van Kampen’s expansion by the fact that weakly damped waves are treated on the same footing as weakly growing waves. It is then found that the free streaming portions of the distribution functions can no longer be left out, and the final result is a system of equations on both the wave amplitudes and the free streaming perturbations, which are coupled through resonant wave particle interaction.
Classification of real simple Lie superalgebras of classical type
Authors: Parker Monique
Published: March 2003

Finite-dimensional simple Lie superalgebras (also called Z2-graded Lie algebras) over an algebraically closed field of characteristic zero were classified in 1976. All simple Lie superalgebras over the reals, whose Lie subalgebra is reductive, are determined here up to isomorphism. As is the theory of simple Lie algebras, this is done by classifying the involutive semimorphisms of the complex Lie superalgebras. One sees in particular that the real form of the Lie subalgebra completely determines the real form of the Lie superalgebra.
Sum rules for zeros of polynomials. I
Authors: Case K. M.
Published: March 2003

It is shown that for polynomials satisfying differential equations of a particular form it is easy to generate sum rules for the powers of the zeros. All of the classical orthogonal polynomials are of this form. Examples are given for the Hermite, Laguerre, Tchebycheff, and Jacobi polynomials. In particular an explicit formula is given for the sums of all powers of Tchebycheff zeros. This same formula gives the sums for general Jacobi polynomials in the limit of large N.
Sum rules for zeros of polynomials. II
Authors: Case K. M.
Published: March 2003

Sum rules for zeros of polynomials which satisfy differential equations with polynomial coefficients of order no higher than the derivative they multiply are extended. First, higher order coefficients are considered. Second, sum rules for negative powers are obtained. Finally, the results are extended to a class of integral functions.
On the separability of the sine-Gordon equation and similar quasilinear partial differential equations. II. Dependent- and independent-variable transformations
Authors: Osborne Anthony D. , Stuart Allan E.G.
Published: March 2003

In an earlier paper, we investigated the separability of the sine-Gordon equation (SGE), and of similar quasilinear partial differential equations, under transformations of the dependent variable (i.e, of the codomain). We found, in particular, that there is a general class of dependent-variable transformations which leads to separable forms of the SGE. In this paper, we extend our previous analysis to include independent- as well as dependent-variable transformations (i.e., transformations of both the domain and codomain) and treat, in detail, constant coefficient equations of the first and second orders. We illustrate our method by applying it to the SGE and find combinations of domain and codomain transformations which reduce the equation to separable forms. Some of these transformations lead to known solutions of the SGE, but others give new solutions expressible in terms of a fifth Painlevé transcendent. Our method can, in principle, be used to map out the space of separable solutions of the SGE and other similar second-order equations, but it does have limitations. A discussion of these limitations and suggestions for possible improvements are also given.
Construction of the phase of the scattering amplitude by iteration of bounds from unitarity integral
Authors: Chan S. K. , Sakmar I. A.
Published: March 2003

Given the modulus of the amplitude a convergent iteration scheme is given to obtain the phase from the unitarity equation. The method is based on improving the upper and lower bounds on the phase. Uniqueness of the solution is investigated and numerical results for a specific case are given.
Particle statistics from induced representations of a local current group
Authors: Goldin G. A. , Menikoff R. , Sharp D. H.
Published: March 2003

Representations of the nonrelativistic current group 𝒮-𝒦 are studied in the Gel’fand–Vilenkin formalism, where 𝒮 is Schwartz’ space of rapidly decreasing functions, and 𝒦 is a group of diffeomorphisms of Rs. For the case of N identical particles, information about particle statistics is contained in a representation of 𝒦F (the stability group of a point F∈𝒮′) which factors through the permutation group SN. Starting from a quasi-invariant measure μ concentrated on a 𝒦 orbit Δ in 𝒮′, together with a suitable representation of 𝒦F for F∈Δ, sufficient conditions are developed for inducing a representation of 𝒮-𝒦. The Hilbert space for the induced representation consists of square-integrable functions on a covering space of Δ, which transform in accordance with a representation of 𝒦F. The Bose and Fermi N-particle representations (on spaces of symmetric or antisymmetric wave functions) are recovered as induced representations. Under the conditions which are assumed, the following results hold: (1) A representation of 𝒮-𝒦 determines a well-defined representation of 𝒦F; (2) equivalent representations of 𝒮-𝒦 determine equivalent representations of 𝒦F; (3) a representation of 𝒦F induces a representation of 𝒮-𝒦; and (4) equivalent representations of 𝒦F determine equivalent induced representations.
Clebsch–Gordan coefficients for corepresentations. II⊗III
Authors: Dirl R.
Published: March 2003

Clebsch–Gordan coefficients for corepresentations are determined quite generally in terms of such coefficients for the normal subgroup, at which the Kronecker products are composed of corepresentations of type II and III.
Quantum inverse method for two-dimensional ice and ferroelectric lattice models
Authors: Thacker H. B.
Published: March 2003

The quantum inverse scattering transform method previously developed for continuum field theories is applied to the exactly soluble symmetric six-vertex (ice or ferroelectric) lattice model. Operators analogous to those which appear in the quantum inverse treatment of the nonlinear Schrödinger and sine–Gordon equations are constructed on the lattice by forming strings of vertices contracted over horizontal arrows. From the commutation relations for these operators, exact formulas for the eigenstates and eigenvalues of the transfer matrix are obtained without making an explicit ansatz for the wave functions. These results illustrate the connection between the quantum inverse method and the transfer matrix formalism for lattice models.
On a class of solutions of the Krook–Tjon–Wu model of the Boltzmann equation
Authors: Barnsley M. , Cornille H.
Published: March 2003

The construction of the solutions of the Krook–Wu model of the nonlinear Boltzmann equation (elastic differential cross sections inversely proportional to the relative speed of the colliding particles) is reduced to the resolution of a nonlinear first order differential system for functions which are related to the moments of the Boltzmann distribution functions. These functions depend upon only one variable, the time, and are the coefficients in the Laguerre expansion of the Tjon-Wu distribution function. We explicitly show how to construct the solutions of the nonlinear differential system and study the properties of the corresponding Tjon-Wu distribution function.
Zero-mass behavior of Feynman amplitudes. I
Authors: Manoukian Edward B.
Published: March 2003

The zero-mass behavior of Feynman amplitudes with or without subtractions is investigated. Rules are given to determine the behavior of any Feynman amplitude, in the most general cases, when some or all of the masses in the underlying theory become small, in general, at different rates in Euclidean space. That we have to consider the approach of such masses to zero at different rates as well is clearly essential on physical grounds. Examples are worked out as illustrations for the rules. A general rule is also given which gives a sufficiency condition to guarantee the existence of the zero-mass behavior of any renormalized Feynman amplitude under the above same conditions which reduces to an elementary and direct inspection of the corresponding Feynman diagram.
A homogeneous Hilbert problem for the Kinnersley–Chitre transformations of electrovac space-times
Authors: Hauser Isidore , Ernst Frederick J.
Published: March 2003

The homogeneous Hilbert problem which we recently formulated for Kinnersley–Chitre transformations of vacuum spacetimes is here generalized to handle transformations of electrovac spacetimes. This provides in particular a simple derivation of our previously published integral equation.
The tensor product of two irreducible representations of the spl(2,1) superalgebra
Authors: Marcu M.
Published: March 2003

The Clebsch–Gordan series and the unnormalized Clebsch–Gordan coefficients for the tensor product of two irreducible representations of spl(2,1) are given. Special emphasis is laid on the discussion of the structure of the series when it contains several nontypical representations. Thereby a better understanding of the ’’degenerate’’ properties of nontypical representations is gained.
Extended phase space. I. Classical fields
Authors: Das A.
Published: March 2003

Classical field theory is developed in the arena of extended phase space V8, the space of position, time, momentum, and energy. This enables one to incorporate Born’s reciprocity which demands equal status for the variables q and p. The present formulation is covariant under the extended Poincaré group P8 acting in V8. Variational methods for classical field theory are generalized. Besides the usual concept of the total 4-momentum, one encounters the notion of average position and time of the field distributions. The total charge emerges from a dynamical viewpoint. The Dirac and Duffin–Kemmer algebras are generalized in this setting. The corresponding wave equations would lead to a dynamical theory of the elementary particles. The symplectic structure is not considered because of the difficulties to represent spinors.
Extended Phase Space. III. Unified spin-1/2 fields
Authors: Das A.
Published: March 2003

In extended phase space V8, the classical field equation for spin-1/2 elementary particles is written as [vk∂/∂qk+ak∂/∂pk−imI] ×ψ(q,p)=0. The 16×16 matrices vk,ak stand for the instantaneous 4-velocity and 4-acceleration. The equation is called the Boltzmann–Dirac–Yukawa (in short BDY) equation. This equation treats q,p variables on equal footings and is covariant under the extended Poincaré group P8. The spin-1/2 field ψ is decomposed into plane wave solutions, and integral constants such as the total energy, the total momentum, the average position, etc., are computed. These integral constants become meaningful provided the modulus squared of each amplitude is interpreted as the statistical distribution function. Next the ψ field is subject to the Hankel transform ψ( ρ,ϑ)∼∑∞t=−∞∑8 R=1∑2α=1∫∞0dκ κ [β(Rα)u(Rα) +ggr̄(Rα)v(Rα)]Jt(κρ)eitϑ=∑∞t=−∞ψ(t); ρ=(q2+p2)1/2, θ=arctan (p/q). The integral constants are constructed from a single (t)-mode ψ(t). These turn out to be physically meaningful for eight spin-1/2 particles. Specially the total charge can be identified with Gell-Man–Nishijima’s expression for the baryon provided the quantum number t3 corresponds to the isotopic spin, 2t4 is identified with the strangeness, and the baryon number b is allowed to take 0,1,2. With each of the (t)-mode ψ(t), eight baryon fields can be associated so that ψ stands for the unified baryon fields. Finally some brief comments are made on the possibility of treating lepton fields within the framework of the BDY equation.
Path-dependent quantum formulation of electromagnetism with magnetic charges
Authors: Gambini R. , Trias A.
Published: March 2003

A path-dependent quantum formulation of electromagnetism with magnetic charges is proposed. One starts from a path-dependent nonsingular potential which generalizes the Wu–Yang potential. Then a Lagrangian formulation is proposed manifestly selfdual and without singular terms or kinematical constraints. The quantization of this Lagrangian leads to a linear space of path-dependent wavefunctions where the dynamical equations are stated. The angular momentum usually considered in the field of a static monopole is identified with the generator of spatial rotations in the space of the path-dependent wavefunctions.
1/N expansion in two-dimensional lattice gauge theory
Authors: Goldschmidt Yadin Y.
Published: March 2003

We use the method of orthonormal polynomials to investigate the large N limit of the two-dimensional lattice gauge theory. The 1/N2 and 1/N4 corrections to the vacuum energy and the expectation value of the Wilson loop operator are calculated in the low temperature phase (λ ≡g2 N<2). These corrections diverge at the transition point. In the high temperature phase we show that there are only corrections which fall exponentially as N becomes large, and hence there are no 1/N corrections in this phase.
Semiclassical results on normalization of bound state wavefunctions
Authors: Moxhay Peter , Rosner Jonathan L.
Published: March 2003

Some methods are introduced and tested for approximating s-wave Schrödinger wavefunctions at the origin, and for examining their dependence on reduced mass of the bound system. Methods are discussed both for nonsingular and singular potentials. The approximations, previously tested for power-law potentials, are found to be exact for the Hulthén potential and excellent for a wide range of parameters in Coulomb+linear potentials: V(r)=−A/r+Br. General theorems for mass dependences are reviewed and extended; these theorems apply to potentials with a definite sign of d2V/dr2. Some specific instances in which d2V/dr2 does not have a unique sign and the theorems do not hold are also discussed.
Analysis of interacting quantum field theory in curved space-time
Authors: Birrell N. D. , Taylor J. G.
Published: March 2003

A detailed analysis of interacting quantized fields propagating in a curved background spacetime is given. Reduction formulas for S-matrix elements in terms of vacuum Green’s functions are derived, special attention being paid to the possibility that the ’’in’’ and ’’out’’ vacuum states may not be equivalent. Green’s functions equations are obtained and a diagrammatic representation for them given, allowing a formal, diagrammatic renormalization to be effected. Coordinate space techniques for showing renormalizability are developed in Minkowski space, for λφ3(4,6) field theories. The extension of these techniques to curved spacetimes is considered. It is shown that the possibility of field theories becoming nonrenormalizable there cannot be ruled out, although, allowing certain modifications to the theory, φ3(4) is proven renormalizable in a large class of spacetimes. Finally particle production from the vacuum by the gravitational field is discussed with particular reference to Schwarzschild spacetime. We shed some light on the nonlocalizability of the production process and on the definition of the S matrix for such processes.
Nondiffractive scattering: Scattering from kaleidoscopes
Authors: Sutherland Bill
Published: March 2003

We introduce and exactly solve a series of multidimensional scattering problems. The potential is a generalization of the delta function potential and has the symmetry of a kaleidoscope. The solution demonstrates the property of nondiffractive scattering. The bound state energies are calculated exactly for all cases. We prove a remarkable identity for certain finite groups.
Gravitational and electromagnetic potentials of the stationary Einstein–Maxwell field equations
Authors: Jones Thaddeus C.
Published: March 2003

We develop the necessary equations for the production of an infinite hierarchy of potentials for stationary Einstein–Maxwell spacetimes. Specific knowledge of these potentials are essential to the Kinnersley–Chitre method of solution generation. To elucidate their meaning we investigate the particular case of Reissner–Nordstrom.
Nonequilibrium fluctuations in driven systems
Authors: Lavenda B. H.
Published: March 2003

The validity of a fluctuation-dissipation theorem for nonequilibrium fluctuations in driven systems is discussed. It is found that the Gaussian assumption alone does not necessarily lead to a fluctuation-dissipation theorem but rather the Markov property must also be invoked. Consequences of the Gaussian–Markov assumption are: (i) the spectral density reduces to a sum of two independent contributions, one due to internal noise and the other to external excitations; and (ii) the mode of regression of nonequilibrium fluctuations is identical to the decay of fluctuations to equilibrium. Without the Markov assumption, time-varying external fields can induce additional correlations which: (i) influence the regression of fluctuations; and (ii) invalidate the fluctuation-dissipation theorem.
Inverse scattering. II. Three dimensions
Authors: Newton Roger G.
Published: March 2003

Assuming that a scattering amplitude, given as a function of the energy and the directions of the incident and scattered particles, is associated with a local potential without spherical symmetry via the Schrödinger equation in three space dimensions, this potential is uniquely reconstructed by two methods. One is based on a generalization of the Marchenko equation; the other, on a generalization of the Gel’fand–Levitan equation.
Addition of complex angular momentum operators
Authors: Joshi K. N. , Rajput B. S.
Published: March 2003

Representations of the components of complex angular momentum operator have been carried out in three different subspaces of the space in which the generators of proper, orthochronous homogeneous Lorentz group operate. Addition of complex angular momentum operators have been derived in the canonical basis of their eigenvectors.
Exact solutions of Brans–Dicke theory for irrotational barotropic models with stiff matter
Authors: Johri V. B. , Goswami G. K.
Published: March 2003

In the previous work, we have obtained classes of exact solutions of Brans–Dicke (BD) theory when only a scalar field is present and also in the presence of an electromagnetic field. In the present paper, we have generated a new class of solutions of BD theory corresponding to irrotational barotropic perfect fluid with pressure equal to energy density. The main result of the paper may be stated as follows: ’’Corresponding to any diagonalizable solution of Einstein vacuum field equations, in which fields and metric tensors are functions of not more than three variables, we can generate a solution of the Brans–Dicke theory for the irrotational barotropic perfect fluid with pressure equal to energy density.’’
Theory of involutional transformations applied to the Dirac theory of the electron. II. Remarks on the Dirac–Coulomb waves
Authors: Kim Shoon K.
Published: March 2003

By means of a single involutional transformation, the Dirac equation for a charged particle in a central Coulomb field is reduced to a second-order differential equation for the radial part which has the same form as the radial equation for the nonrelativistic Coulomb problem. The general Dirac–Coulomb waves thus obtained are highly symmetric with respect to signs of the energies and other constants of motion. It gives the precise phase shift for the continuum spectrum without any additive multiple of π in contrast to the existing solutions. The symmetry properties of the Dirac–Coulomb waves are discussed.
The existence of wave operators for oscillating potentials
Authors: Devinatz Allen
Published: March 2003

The existence of the Mo/ller wave operators is proved for Hamiltonians of the form H=−Δ+a sinbrα/rβ+V(x), where V is a short range potential, generally noncentral, and α and β take on suitable values including the case α=1, β≳ 1/2 .
Fock–Tani representation for composite particles in a soluble model
Authors: Girardeau M. D.
Published: March 2003

The transformation from the usual Fock representation to the Fock–Tani representation for composite particles is carried out for a simplified model of the composite which allows closed-form expressions for the transformed quantities. A ’’statistical renormalization’’, in which single particle energies and interactions become dependent on the composite particle occupation number, plays an essential rôle in the solution, allowing absorption of unlinked terms into the definition of the renormalized energies.
The isotropic harmonic oscillator in an angular momemtum basis: An algebraic formulation
Authors: Bracken A. J. , Leemon H. I.
Published: March 2003

A completely algebraic and representation-independent solution is presented of the simultaneous eigenvalue problem for H, L2, and L3, where H is the Hamiltonian operator for the three-dimensional, isotropic harmonic oscillator, and L is its angular momemtum vector. It is shown that H can be written in the form h/ω(2ν°ν+λ°⋅λ+3/2), where ν° and ν are raising and lowering (boson) operators for ν°ν, which has nonnegative integer eigenvalues k; and λ° and λ are raising and lowering operators for λ°⋅λ, which has nonnegative integer eigenvalues l, the total angular momentum quantum number. Thus the eigenvalues of H appear in the familiar form h/ω(2k+l+3/2), previously obtained only by working in the coordinating the operators ν° and λ° to a ’’vacuum’’ vector on which ν and λ vanish. The Lie algebra so(2,1) ⊕ so(3,2) is shown to be a spectrum-generating algebra for this problem. It is suggested that coherent angular momentum states can be defined for the oscillator, as the eigenvectors of the lowering operators ν and λ. A brief discussion is given of the classical counterparts of ν,ν°, λ, and λ°, in order to clarify their physical interpretation.
Asymptotic distribution of eigenvalues for the multidimensional Schrödinger equation
Authors: Newell G. F.
Published: March 2003

Let N(λ) be the number of eigenvalues less than λ of the k-dimensional Schrödinger equation for which the potential energy becomes infinite as x→∞ in all directions of the k-dimensional space. Titchmarsh has derived a first order asymptotic approximation to N(λ) for λ→∞. His formula is rederived here by a different method and then extended so as to give at least one more term of the asymptotic series. The method is based upon approximating q(x) locally by a quadratic function and evaluating the short time properties of the propagator for a hypothetical time-dependent diffusion equation having the same eigenvalues.
Shift operator techniques for the classification of multipole-phonon states: IV. Properties of shift operators in the G2 group
Authors: Vanden Berghe G. , Meyer H. E. De
Published: March 2003

A previously developed shift operator method is applied to the G2 group, which plays an important role in the classification of nuclear octupole-phonon states. Expressions which connect quadratic products of the considered shift operators with G2 invariants are derived.
Shift operator techniques for the classification of multipole-phonon states. V. Properties of shift operators in the R(7) group
Authors: De Meyer H. E. , Vanden Berghe G.
Published: March 2003

With a view to obtaining additional label-generating operators for the classification of octupole-phonon states a set of shift operators Okl (k=0, ±1, ±2, ±3) in the R(7) group is constructed. Expressions which connect quadratic products of these shift operators are given, and it turns out that besides R(7) invariants the expressions also involve the scalar G2 shift operator P0l previously studied. The opportunity to arrive at an orthogonal solution of the state labelling problem is discussed.
Selection rules for type II Shubnikov space groups
Authors: Dirl R.
Published: March 2003

Clebsch–Gordan coefficients for type II Shubnikov space groups, containing the inversion as point group operation are expressed by simple formulas in terms of convenient Clebsch–Gordan coefficients for the unitary subgroup.
Congruence number, a generalization of SU(3) triality
Authors: Lemire F. , Patera J.
Published: March 2003

Congruence classes of finite-dimensional representations of semisimple Lie groups are defined. Each class is characterized by a congruence number. For the group SU(3) the concept reduces to the familiar triality number.
Partial inner product spaces. IV. Topological considerations
Authors: Antoine J.-P.
Published: March 2003

Whereas the third paper in this series dealt with the algebraic structure of partial inner product (PIP) spaces, the present one explores systematically their topological properties. A slightly more restricted object is introduced, that we call an indexed PIP-space: it consists of a PIP-space together with a distinguished family of assaying subspaces. The upshot of the analysis is the characterization of two types of indexed PIP-spaces, called type (B) and type (H), respectively, as the most likely candidates for practical applications; they are simply lattices of Banach, resp. Hilbert spaces. Operators on indexed PIP-spaces are discussed and conditions are given that guarantee that the domain of any such operator is a vector subspace. Finally, we examine the question of the existence of a central Hilbert space, in the case of a positive definite partial inner product.
Nth-order multifrequency coherence functions: A functional path integral approach. II
Authors: Rose Conrad M. , Besieris Ioannis M.
Published: March 2003

Nth-order multifrequency coherence functions arising in beam propagation through focusing media with random-axis misalignments and focusing media with additive statistical fluctuations are computed. The analysis is carried out by means of a simple formula which yields exact algorithmic solutions to a class of canonical path integrals.
A simple approximate determination of stochastic transition for the standard mapping
Authors: Mortell Michael P. , Seymour Brian R.
Published: March 2003

The standard mapping results from a study of nonlinear forced oscillations of a gas in a closed tube. On interpreting stochastic transition as the breaking of all waves during a round trip in the tube, an estimate for the critical value of the relevant parameter is simply derived. The estimate agrees very well with the result of a more elaborate analysis of Greene.
Image methods for constructing Green’s functions and eigenfunctions for domains with plane boundaries
Authors: Terras Riho , Swanson Robert
Published: March 2003

We consider the image problem for domains with plane boundaries. We list all three and two dimensional domains for which the image method yields solutions of the potential problem, and we describe the image arrays generated by these domains in familiar crystallographic terms. One obtains from the group–theoretic description of images two representations for the Dirichlet Green’s functions for ∇2. The first is obtained by summing the unrestricted Green’s function over the crystal image structures, and the second is obtained in terms of an eigenfunction expansion using solutions of ∇2ψ=λψ which vanish on the plane boundaries.
Transition amplitudes for time-dependent harmonic oscillators
Authors: Landovitz L. F. , Levine A. M. , Schreiber W. M.
Published: March 2003

Utilizing the Green’s function for a time dependent harmonic oscillator, we calculate the corresponding transition amplitudes. Particular examples of damped and runaway oscillators are discussed.
Thermodynamics. II. The extended thermodynamic system
Authors: Zeleznik Frank J.
Published: March 2003

The algebraic theory of thermodynamics developed in a previous paper is extended to include the algebraic structure that arises from the introduction of a physical body into the theory. The extension is based on very general definitions of both the thermodynamic states of a body and subsystems of that body. The algebraic analysis, which includes bodies in nonuniform states, shows that the set of all thermodynamic states of a body has the same algebraic structure as the set of thermodynamic states and that composite systems are induced by the algebraic structure of thermodynamic states. The analysis also justifies a variational treatment of thermodynamic bodies in uniform as well as nonuniform states. The variational calculation includes all conventional methods of calculation as special cases and helps to illuminate the origin and interpretation of the electrochemical potential.
SU(m
n⊇SU(m)×SU(n) isoscalar factors and S(f1+f2)⊇S( f1)×S( f2) isoscalar factors
Authors: Chen Jin-Quan
Published: March 2003

A simple relation is found between the isoscalar factor (ISF) of the unitary group and those of the permutation group, i.e. the SU(mn)⊆SU(m)×SU(n)ISF is equal to the S( f1+f2)⊆S( f1)×S( f2) ISF. Since the values of S( f1+f2)⊆S( f1)×S( f2)ISF are independent of m and n, one arrives at an important conclusion that the values of SU(mn)⊆SU(m)×SU(n) ISF are also independent of m and n. Therefore they can be calculated for all m and n by a single stroke instead of one m and one n at a time. An eigenfunction metho for evaluating the SU(mn)⊆SU(m)×SU(n)ISF is given which can be easily translated into a computer program.
On the WKB approximation to the propagator for arbitrary Hamiltonians
Authors: Mizrahi Maurice M.
Published: March 2003

This paper presents a general expression for the WKB approximation to the propagator corresponding to an arbitrary Hamiltonian operator H. For example, if the correspondence rule used to pass from the classical Hamiltonian Hc to H is such that it associates aPiQ j +(1−a)Q jPi to piq j, then the formula gives KWKB=KVVexp {( 1/2 −a)ℱT(∂2Hc/∂qi∂pi )(qc(t),pc(t),t) dt}, where KVV≡(2πih/)−n/2 (det M)1/2exp(iSc/h/) is Van Vleck’s well-known formula, Sc being the action functional evaluated at the classical path (qc,pc) and Mij ≡−∂2Sc/∂qa i∂qbj. More generally, the formula presented here applies to any system with n degrees of freedom described by a function f(x,t) whose time evolution is given by (H(x,k∂/∂x,t)+k∂/∂t) f(x,t)=0, regardless of the form of H. The Schrödinger equation of quantum mechanics and the Fokker–Planck equation of diffusion are obvious examples. Many examples are discussed. This generalizes results obtained in a previous publication [J. Math. Phys. 18, 786–90 (1977)].
Solution of the inverse problem in multigroup transport theory
Authors: Larsen Edward W.
Published: March 2003

A solution is given for the (inverse) problem of determining the scattering laws for a multigroup, anisotropically scattering medium, in terms of the boundary fluxes corresponding to finite slabs.
Reduction technique for matrix nonlinear evolution equations solvable by the spectral transform
Authors: Calogero F. , Degasperis A.
Published: March 2003

The main purpose of this paper is to describe a technique of reduction, whereby from the class of evolution equations for matrices of order N solvable via the spectral transform associated to the (matrix) linear Schrödinger eigenvalue problem, one derives subclasses of nonlinear evolution equations involving less than N2 fields. To illustrate the method, from the equations for matrices of order 2 two subclasses of equations for 2 fields (rather than 4) are obtained. The first class coincides, or rather includes, that solvable via the spectral transform associated to the generalized Zakharov–Shabat spectral problem; further reduction to nonlinear evolution equations for a single field reproduces a number of well-known equations, but also yields a novel one (highly nonlinear). The second class also yields highly nonlinear equations; some examples are given, including another novel evolution equation for a single field.
Nonorthogonal R-separable coordinates for four-dimensional complex Riemannian spaces
Authors: Kalnins E. G. , Miller W.
Published: March 2003

We classify all R-separable coordinate systems for the equations Δ4Ψ=𝒥4i, j=1g−1/2 ∂ j(g1/2gi j∂iΨ) =0 and 𝒥4i, j=1gi j∂iW∂ jW =0 with special emphasis on nonorthogonal coordinates, and give a group theoretic interpretation of the results. For flat space we show that the two equations separate in exactly the same coordinate systems and present a detailed list of the possibilities. We demonstrate that every R-separable system for the Laplace equation Δ4Ψ=0 on a conformally flat space corresponds to a separable system for the Helmholtz equations Δ4Φ=λΦ on one of the manifolds E4, S1×S3, S2×S2, and S4.
Boson, para-boson, and boson-fermion representations of some graded Lie algebras
Authors: Jagannathan R. , Vasudevan R.
Published: March 2003

Some graded Lie algebras with the Lie algebras so(3), so(2,1), so(4), so(3,1), and so(2,2) as their Bose sectors are realized in terms of bosons, para-bosons and certain bilinear combinations of bosons and fermions.
Extension of inverse scattering method to nonlinear evolution equation in nonuniform medium
Authors: Gupta M. R. , Ray Jayanti
Published: March 2003

By allowing the entire spectrum of certain linear eigenvalue problems to evolve with time a general type of nonlinear evolution equation in nonuniform medium which is exactly integrable by the inverse scattering method has been derived. The derivative nonlinear Schrödinger equation or the nonlinear Schrödinger equation with linear or parabolic density profiles are special cases of this generalized form.
Classical mechanics, the diffusion (heat) equation and the Schrödinger equation on a Riemannian manifold
Authors: Elworthy David , Truman Aubrey
Published: March 2003

We consider the limiting case λ→0 of the Cauchy problem, ∂gλ(x,t)/∂t = (1/2) λΔxgλ(x,t)+(V(x)/λ) gλ( x,t), with gλ (x,0) = exp{−S0(x)/λ}T0(x), V, S0 being real-valued functions on N, T0 a complex-valued function on N; V, S0, T0 being independent of λ, Δx being the Laplace–Beltrami operator on N, some complete Riemannian manifold. We prove some new results relating the limiting behavior of the solution to the above Cauchy problem to the solution of the corresponding classical mechanical problem  D2Z(s)/∂s2 = −∇ZV[Z(s)], s∈[0,t], with Z(t) = x and Z(0) = ∇S0(Z(0)).One of our results is equivalent to the fact that for short times Schrödinger quantum mechanics on the Riemannian manifold N tends to classical Newtonian mechanics on N as h/ tends to zero.
On rotations in a pseudo-Euclidean space and proper Lorentz transformations
Authors: Rao K. N. Srinivasa , Saroja D. , Rao A. V. Gopala
Published: March 2003

It is shown that in a general pseudo-Euclidean space Epn, 2-flats (planes) passing through the origin of the coordinate system may be classified into six invariant types and explicit formulas for ’’planar rotations’’ in these flats are obtained. In the physically important case of the Minkowski World E34, planar rotations are characterized as rotationlike, boostlike, and singular transformations and an invariant classification of proper Lorentz transformations into these types is given. It is shown that a general nonsingular proper Lorentz transformation may be resolved as a commuting product of two transformations one of which is rotationlike and the other boostlike while a singular transformation may be written as a product of two rotationlike transformations, each with a rotation angle π. Such a rotationlike transformation with angle π is called ’’exceptional’’ following Weyl’s terminology for similar transformations of SO(3). In all cases, explicit formulas for the angles and planes of rotations in terms of the elements of a given Lorentz matrix are obtained and the procedure yields in a natural manner an explicit formula for the image of L in the D10(D01) representation of SO(3,1) which in turn leads to two more classification schemes in terms of the character χ of L in the D10(D01) and the D (1)/(2) 0(D0 (1)/(2) ) representations.
A new direct proof of the expansion of a Coulomb-distorted plane wave in Coulomb-distorted spherical waves
Authors: Furukawa Masao
Published: March 2003

By employing specific properties of confluent hypergeometric functions, it has been directly proved that a Coulomb-distorted plane wave is expressible in superposition of Coulomb-distorted spherical waves.
The Palatini variational principle for the general Bergmann–Wagoner–Nordtvedt theory of gravitation
Authors: Van den Bergh N.
Published: March 2003

The Hilbert variational principle for the Bergmann–Wagoner–Nordtvedt scalar–tensor theory of gravitation is presented in a general form, yielding the field equations in arbitrary units. Applying then the Palatini method of variation to this action integral, the connection can be shown to be an integrable Weyl connection, even if the matter part contains explicitly the connection coefficients. However, the resulting field equations always appear in the wrong unit system, or turn out to be wrong in a general gauge.
Bounds on Green’s functions of second-order differential equations
Authors: van den Berg M.
Published: March 2003

We estimate the diagonal part of the Green’s function for the equation (−Δ/2+V(x)+∂/∂t)ψ(x, t) = 0, t≳0, x∈ B, where B is a finite region of the Euclidean space Rd with a regular boundary. In the special case V(x) = 0, x∈B, we also obtain bounds for the nondiagonal part of the Green’s function which are uniform in t.
The Coulomb unitarity relation and some series of products of three Legendre functions
Authors: van Haeringen H. , Kok L. P.
Published: March 2003

We obtain from the off-shell Coulomb unitarity relation a closed expression for 𝒥∞l = 0(2l+1)Pl(x) ×Qliγ ( y) Ql−iγ (z), and we consider some related series of products of Legendre functions.
Representation theory of compact groups over fields of characteristic zero
Authors: Kasperkovitz P.
Published: March 2003

The group algebras over the reals, the complex numbers, and the quaternions, are discussed and compared for a fixed compact group. Special emphasis is put on the properties of ring bases of the minimal two-sided ideals and their relation to the matrix representation of the group irreducible over the fields cited above.
Bäcklund transformations connecting different isospectral deformation equations
Authors: Chudnovsky D. V. , Chudnovsky G. V.
Published: March 2003

We study singular spectral problems on Riemann surfaces and isospectral deformation equations connected with them. We show that for a given class of reductions all spectral problems on a given Riemann surface give rise to the same class of isospectral deformation equations. These isospectral deformation equations can be solved using the Riemann boundary value problem. All possess local conservation laws and Bäcklund transformations.
Inverse problem for the reduced wave equation with fixed incident wave. II
Authors: Weston V. H.
Published: March 2003

The reduced wave equation Δu+k2n2(x)u = 0 for x in R3 is considered where [n2−1] is a real function of compact support given by the domain D. Sources outside D produce an incident field ui (satisfying the equation with n≡1 for all x). This generates a scattered field us(x,n) satisfying the reduced wave equation and the radiation condition. The scattered field is measured at a set of points {xj}Nj = 1 outside D. If n* is a known quantity for which us(x,n*) can be computed, the inverse problem considered here, consists of solving the nonlinear system of functional equations us(xj,n) −us(xj,n*) = bj, j = 1,2,...,N for an n(x) such that ℱD (n2−n2*)2W dx is minimized. With restrictions on the data set {bj}Nj = 1, a unique solution can be generated in a certain ball by iterating a system of nonlinear integral equations. It is shown that n(x), with less stringent conditions than that of continuity, may be treated.
On a model of a harmonic oscillator coupled to a quantized massless, scalar field. II
Authors: Arai Asao
Published: March 2003

In terms of the resonance pole of the S matrix, which is also a complex pole associated with the 2-point function for the harmonic oscillator, a rigorous mathematical interpretation is given to the formal perturbation calculation for the ’’level shifts’’ (the Lamb shifts in QED) of the harmonic oscillator atom and for the ’’decay probabilities’’ of the excited states due to the spontaneous emission of bosons.
Generalized monotone convergence and Radon–Nikodym theorems
Authors: Gudder S. , Zerbe J.
Published: March 2003

A measure and integration theory is presented in the quantum logic framework. A generalization of the monotone convergence theorem is proved. Counterexamples are used to show that the dominated convergence theorem, Fatou’s lemma, Egoroff’s theorem, and the additivity of the integral do not hold in this framework. Finally, a generalization of the Radon–Nikodym theorem is proved.
Stochastic quantization of the electromagnetic field
Authors: Davidson Mark
Published: March 2003

Stochastic field equations for electromagnetism are presented. The covariance in the ground state is calculated. The question of Lorentz covariance is examined. The connection with ordinary quantum field theory is made. The theory is compared with stochastic electrodynamics.
Trajectories of the S-matrix poles in the complex wavenumber plane
Authors: Sanz P. , Sañudo J. , Sesma J.
Published: March 2003

The singularities of the S matrix for a square potential, and their trajectories in the complex wavenumber plane as the angular momentum varies through continuous real values, are presented. Some enlightenment about the physical interpretation of Regge trajectories is obtained.