bluuebunny/crossref_metadata_2025 huggingface dataset:
Solution of potential problems near the corner of a conductor
Authors: Davis L. C. , Reitz J. R.
Published: March 2003

The Green’s function for a space defined (in cylindrical coordinates) by the intersection of two half-planes S1 (φ=0) and S2 (φ=ϑ where 0 < ϑ ⩽ 2π) is found by a technique due to Sommerfeld. The Green’s function (or its normal derivative) is required to vanish on the surface S1 + S2 as well as at infinity. When ϑ = mπ/k where k and m are integers, the solution can be written in terms of the Green’s function um for a Riemann space of m windings (in φ). For m = 1 and 2, um can be expressed in terms of elementary functions. For m = 3, we find um to be given in terms of complete elliptic integrals. Application to some simple electrostatic and magnetostatic problems is made, particularly for ϑ = 3π/2.
The de Donder coordinate condition and minimal class 1 space–time
Authors: Krause Jorge
Published: March 2003

By means of immersion techniques a set of ’’adapted coordinates’’ are introduced as preferred coordinates for class 1 space–time. It is proved that the necessary and sufficient condition for the adapted coordinates to be harmonic coordinates is that class 1 space–time be a minimal variety. Some interesting features of the embedding approach to curved space–time are also shown in terms adapted coordinates.
Two applications of the Racah coefficients of the Poincaré group
Authors: Klink W. H.
Published: March 2003

Racah coefficients of the Poincaré group are defined and compared with the rotation group. It is then shown how they arise in a natural manner in the inelastic unitarity equations and in crossing multiparticle amplitudes.
General chiral S
U2 × S
U2 pion Lagrangian and the generators of O (4,1) group
Authors: Ebrahim Abdul
Published: March 2003

It is shown that the general chiral SU2 × SU2 invariant pion Lagrangian in the form given by Gürsey can be used to obtain a general and convenient parametrization of an infinite set of generators of the noncompact O (4,1) group. Various special cases of the general form of the generators are given, and one particular form is shown to coincide with the generators of the O (4,1) group used in the literature.
Optical theorems for three-to-three processes
Authors: Coster Joseph , Stapp Henry P.
Published: March 2003

Optical theorems for three-to-three processes are derived from S-matrix principles. These theorems express all single, double, and multiple discontinuities across all combinations of normal threshold cuts in terms of physical scattering amplitudes. The 216 functions corresponding to all combinations of sides of the 16 normal threshold cuts are determined by analyticity requirements and the generalized Steinmann relations. These two conditions guarantee that these functions can be identified with the corresponding functions in the Regge discontinuity formulas of Weis. This identification provides for a possible enlargement of the scope of Regge–Mueller-type analyses of high-energy processes.
Minimal coupling and complex line bundles
Authors: Greub Werner , Petry Herbert-Rainer
Published: March 2003

The concept of minimal coupling, which leads to the Schrödinger equation of a particle in an external electromagnetic field, is reformulated within the theory of complex line bundles. The possible generalizations are discussed, and the case of the magnetic monopole is investigated with the help of the new formalism.
Addendum: Determination of the amplitude from the differential cross section by unitarity
Authors: Newton Roger G.
Published: March 2003

JMAPA,9,2050,1968
Orthogonal polynomials. II
Authors: Case K. M.
Published: March 2003

A class of orthogonal polynomials defined by a weight function of compact support is considered. These are known to satisfy three-term recursion relations. It is shown, under rather weak restrictions, that the traces of powers of the Jacobi matrices formed from the coefficients in the recursion relations are simply related to Fourier coefficients of the logarithm of the weight function.
Separation of variables in the Hamilton–Jacobi, Schrödinger, and related equations. I. Complete separation
Authors: Havas Peter
Published: March 2003

It was established by Levi-Civita that in n dimensions there exist n+1 types of coordinate systems in which the Hamilton–Jacobi equation is separable, n of which are in general nonorthogonal; the form of the separated equations was given by Burgatti and Dall’Acqua. In this paper first the general forms of the n+1 types of metric tensors of the corresponding corresponding Riemannian spaces Vn are determined. Then, sufficient conditions are given for coordinate systems in which the Schrödinger, Helmholtz, and Laplace equation are separable. It is shown that there again exist n+1 types of such systems, whose metric tensors are of the same form as those of the Hamilton–Jacobi equation. However, except for the ’’essentially geodesic case’’ of Levi-Civita they are further restricted by a condition on the determinant of the metric; this condition is a generalization of that found by Robertson for orthogonal systems in the case of the Schrödinger equation.
Second quantization of classical nonlinear relativistic field theory. I. Canonical formalism
Authors: Bal/aban T. , Ra̧czka R.
Published: March 2003

The canonical formalism for nonlinear relativistic classical field theory is presented. It is shown that the solutions Φ (x) of the nonlinear equation (⧠+m2) Φ (x) =λΦ3(x) as well as the asymptotic fields Φin(x) and Φout(x) are local relativistic fields with respect to Poisson brackets, with initial data as canonical variables. A convenient form for the generators of the Poincaré group is derived, and the properties of the scattering operator are discussed.
Spectrum of the Kirkwood–Salsburg operator and phase transitions
Authors: Klein William
Published: March 2003

We prove that the spectrum of the Kirkwood–Salsburg equation in finite volumes is composed only of eigenvalues.
Canonical realizations of the Poincaré group. I. General theory
Authors: Pauri M. , Prosperi G. M.
Published: March 2003

The canonical realizations of the full Poincaré group in classical mechanics are studied by means of a general formalism introduced in preceding papers. The resulting classification displays significant analogies with the quantum one. The irreducible realizations, corresponding to positive, zero, and imaginary mass particles with or without spin, are discussed. Also the irreducible realizations of the homogeneous Lorentz group are classified. Particular attention is given to the nonirreducible realization describing a system of two free particles and to a discussion of the physical meaning of the ’’center-of-mass’’ and ’’internal’’ variables. It is seen that this formalism provides a most natural framework for the introduction of a direct interaction between the particles according to the well-known prescription given by Bakamjian and Thomas. Finally, some simple models of relativistic ’’rigid’’ systems are discussed.
Boundedness of linear integral operators in weighted L
p spaces
Authors: Ronen Yigal
Published: March 2003

Boundedness properties play a role in the solutions of integral equations as well as in the question of error estimations. A test for the boundedness of linear integral operators in weighted Lp spaces is presented, and applied to the neutron energy dependent transport operator.
The triad-interaction representation of homogeneous turbulence
Authors: Lee Jon
Published: March 2003

The triad-interaction representation has been presented for the 2D and 3D homogeneous flows. This has several advantages over the usual Fourier-amplitude representation: (i) The incompressibility is built into the equation as in the vorticity equation. (ii) For a given wave vector, the number of dynamic equations is one less than that of the Fourier-amplitude equations. (iii) In the inviscid limit, energy and enstrophy are conserved in 2D, whereas the 3D flow conserves energy and helicity. (iv) The entire family of triad interactions is categorized into two classes in 2D and four classes in 3D, according to the geometry of triad wave vectors. Lastly, (v) the necessary conditions for isotropy in 3D emerge as the reflexional, rotational, and spherical symmetries in the wave vector space, whereas polar symmetry is only the requirement in 2D. The triad-interaction representation has proved very useful in the investigation of isolating constants of motion and the statistical theory of nonisotropic turbulence.
An invariance theory for second-order variational problems
Authors: Logan J. D. , Blakeslee J. S.
Published: March 2003

This paper investigates the invariance properties of second-order variational problems when the configuration space is subjected to an r-parameter local Lie group of transformations. In particular, the recent results of Hanno Rund on first-order problems are extended to the higher order case: A new set of fundamental invariance identities are derived for single and multiple integral problems, and new proofs of the Zermelo conditions and Noether’s theorem are presented. The results are applied to a variational problem whose second-order Lagrangian depends upon a scalar field in Minkowski space, and some conformal identities are obtained.
Simplified calculations for radiation reaction forces
Authors: Tabensky R. , Villarroel D.
Published: March 2003

The Lorentz–Dirac equation of motion for the electron is derived by a new method which makes tedious power series expansions unnecessary.
Analysis of the dispersion of low frequency uniaxial waves in heterogeneous periodic elastic media
Authors: Balanis George N.
Published: March 2003

The dispersion of harmonic uniaxial waves in heterogeneous periodic elastic media is investigated. The frequency dependence of the wave phase velocity is obtained in the form of a power series valid for small frequencies and arbitrary spatial variations of the heterogeneities. The dominant dispersion term is always negative and proportional to the square of the frequency. Near the static limit of zero frequency the dispersion is of the normal type—the group velocity, which is also a quadratic decreasing function of the frequency, decreases faster than the phase velocity.
The Hamiltonian H = (−1/2) d2/d
x2 + x2/2 + λ/x2 reobserved
Authors: Lathouwers L.
Published: March 2003

The Schrödinger problem for the title Hamiltonian is considered as a perturbed one-dimensional harmonic oscillator. Exact bound state solutions can be derived from a classical differential equation in the theory of Laguerre polynomials. These solutions are valid and analytically dependent on λ only in a limited range of the perturbation strength. Within this region the oscillator Hamiltonian restricted to odd and even parity subspaces is unitary equivalent to H restricted over the respective perturbed subspaces. It is shown that due to the singular nature of the perturbation the allowed λ range is narrowed if side conditions are imposed to make the wavefunctions ’’physically interpretable.’’
On convergent iterations and their applications to Hartreee–Fock equations
Authors: Kelemen A.
Published: March 2003

A method for solving a restricted class of nonlinear equations is presented and applied in detail to solution of the Hartree–Fock (HF) equations.
On the projective unitary representations of connected Lie groups
Authors: Cariñena Jose F. , Santander Mariano
Published: March 2003

We show here the possibility of finding a unique Lie group Ḡ associated with each connected Lie group G such that every projective unitary representation of Ḡ can be lifted to a unitary representation of Ḡ, that is to say, all PUR of G can be found from the UR of only one group Ḡ. This method is applied to the research of PUR of the Galilei group and compared with the preceding ones.
Neutron transport and diffusion in inhomogeneous media. I
Authors: Larsen Edward W.
Published: March 2003

The asymptotic solution of the neutron transport equation is obtained for large near-critical domains D which possess a cellular, nearly periodic structure. A typical mean free path in D is taken to be of the same order of magnitude as a cell diameter, and these are taken to be small (of order ε) compared to a typical diameter of D. The solution is asymptotic with respect to the small parameter ε. It is a product of two functions, one determined by a detailed cell calculation and the other obtained as the solution of a time dependent diffusion equation. The diffusion equation contains precursor (delayed neutron) densities, equations for which are derived. The coefficients in the diffusion equation, which are determined using the results of the cell calculation, differ from those now used in engineering applications. The initial condition for the diffusion equation is derived, and the problem of determining the boundary condition is discussed.
On the symmetries of the Racah coefficient
Authors: Rao K. Srinivasa , Santhanam T. S. , Venkatesh K.
Published: March 2003

Observations on the symmetries of the Racah coefficient are made in terms of its three allowed series representations. The Bailey’s transform between the two terminating Saalschutzian 4F3(1) series, when applied to the 4F3(1) representation for the Racah coefficient, is shown to result (at best) in the allowed substitution j → −j −1 for any one of the six angular momenta in the Racah coefficient.
Construction of a meromorphic many-channel p-wave S matrix
Authors: Cox Joseph R. , Garcia Hugo R.
Published: March 2003

The simplest possible nonrelativistic many-channel p-wave S matrix which is meromorphic on its energy Riemann surface, as well as the underlying potential matrix, is explicitly constructed by means of the many-channel Marchenko equations. The results suggest that, contrary to the case in which no coupling between channels is present, such an S matrix necessarily increases in complexity with increasing angular momentum.
The symmetrical energy–momentum tensor derived by parametrization
Authors: Hjalmars Stig
Published: March 2003

With reference to a paper by Goedecke in this journal attention is drawn to the fact that already in his original paper on the subject Rosenfeld proved the equality of the results of the two general procedures of symmetrizing energy–momentum tensors, i.e., the procedure of Belinfante (1939), utilizing the angular–momentum tensor, and the procedure of Rosenfeld (1940), taking the Lorentz metric limit of the manifestly symmetrical energy–momentum tensor of Riemannian space. Since Rosenfeld’s presentation of his procedure may give the misleading impression that it has something to do with curved spaces, general relativity, or gravitational theory, we show in the present paper how his scheme can be recast in a form, where one merely takes resort to an infinitesimal transformation of the ordinary Lorentz coordinates to arbitrary curvilinear coordinates, describing the same original Lorentz space of zero curvature. This transformation, of course, means a parametrization of the variational principle, and the analysis can thus be performed by means of a generalization of the theory of parameter-invariant variational principles. An expression for the symmetrized energy–momentum tensor is given, which is equivalent to that given by Rosenfeld, and in which the transformation functions are seen to vanish identically. The procedure is thus seen to be not so much a limiting process as a transformation to curvilinear coordinates, construction of a symmetrical energy–momentum tensor, and a transformation back again.
Irreducible Cartesian tensor expansions of scalar fields
Authors: Moraal H.
Published: March 2003

It is shown how a scalar function V (‖R + Σni=1 ai‖) of a sum of n + 1 vectors can be expanded as a multiple Cartesian tensor series in the vectors ai. This expansion is a rearrangement of the multiple Taylor series expansion of such a function. In order to prove the fundamental theorem, Eq. (3.1) below, generalized Cartesian Legendre polynomials are defined. The theorem is applied to the eigenfunctions of the Laplace operator and to inverse powers. The expansions of the latter type of function leads to forms involving generalized hypergeometric functions in several variables. As a special case, the Cartesian form of the multipole expansion of the electrostatic potential between two linear molecules is derived. A number of sum rules for hypergeometric functions and addition formulas for (standard and modified) spherical Bessel functions are proved using a reduction property of the generalized Legendre polynomials. The case of the expansion of a tensorial function is also briefly discussed.
Onsager symmetry and the diagonalizability of the hydrodynamic matrix
Authors: Maltman Kim , Laidlaw W. G.
Published: March 2003

The diagonalizability of the hydrodynamic matrix in the case that one of the variables is odd under time reversal is investigated. The implications for a normal mode analysis and for the spectral density elements are considered.
A nonlinear system of Euler–Lagrange equations. Reduction to the Korteweg–de Vries equation and periodic solutions
Authors: Zielke Walter
Published: March 2003

The Euler–Lagrange equations, which correspond to a variational

The Euler–Lagrange equations, which correspond to a variational principle with a Lagrange function depending on arbitrary functions and their first order derivatives, are shown to be reducible to the and their first order derivatives, are shown to be reducible to the Korteweg–de Vries equation under a small—but finite—amplitude approximation. Closed form approximation. Closed form periodic solutions to the Euler–Lagrange equations are found for a particular case, and the modulational stability of these solutions is discussed. Equations for waves in cold plasma are discussed as examples.
A general setting for Casimir invariants
Authors: Abellanas L. , Martinez Alonso L.
Published: March 2003

This paper contains a general description of the theory of invariants under the adjoint action of a given finite-dimensional complex Lie algebra G, with special emphasis on polynomial and rational invariants. The familiar ’’Casimir’’ invariants are identified with the polynomial invariants in the enveloping algebra 𝒰 (G). More general structures (quotient fields) are required in order to investigate rational invariants. Some useful criteria for G having only polynomial or rational invariants are given. Moreover, in most of the physically relevant Lie algebras the exact computation of the maximal number of algebraically independent invariants turns out to be very easy. It reduces to finding the rank of a finite matrix. We apply the general method to some typical examples.
Continuous subgroups of the fundamental groups of physics. II. The similitude group
Authors: Patera J. , Winternitz P. , Zassenhaus H.
Published: March 2003

All subalgebras of the similitude algebra (the algebra of the Poincaré group extended by dilatations) are classified into conjugacy classes under transformations of the similitude group. Use is made of the classification of all subalgebras of the Poincaré algebra, carried out in a previous article. The results are presented in tables listing representatives of each class and their basic properties.
Structural equations for Killing tensors of order two. II
Authors: Hauser I. , Malhiot R. J.
Published: March 2003

In a preceding paper, a new form of the structural equations for any Killing tensor of order two were derived; these equations constitute a system analogous to the Killing vector equations ∇α Kβ = ωαβ = −ωβα and ∇γ ωαβ = Rαβγδ Kδ. The first integrability condition for the Killing tensor structural equations is now derived. Our structural equations and the integrability condition have forms which can readily be expressed in terms of a null tetrad to furnish a Killing tensor parallel of the Newman–Penrose equations; this is briefly described. The integrability condition implies the new result, for any given space–time, that the dimension of the set of second order Killing tensors attains its maximum possible value of 50 only if the space–time is of constant curvature. Potential applications of the structural equations are discussed.
Boundedness below for fermion model theories. I
Authors: Brydges David
Published: March 2003

The spatially cut-off Hamiltonians for the models (ψ̄ψ̄φ)2 and (ψ̄ψ̄φN + φ2M)2 with M ≳ N are bounded below uniformly in a momentum cutoff, by using the semi-Euclidean formulation.
Foundations of a quantum probability theory
Authors: Srinivas M. D.
Published: March 2003

Statistical physical theories are frequently formulated in terms of probabilistic structures founded on a ’’logic of experimentally verifiable propositions.’’ It is argued that to each experimentally verifiable proposition there corresponds an experimental procedure which, in general, alters the state of the system, and is completely characterized by a ’’measurement transformation’’ or ’’operation.’’ An analysis of the relations among these experimental procedures leads us to a ’’logic of operations’’ which is quite different from the ’’lattice theoretic logics’’ that are often considered (albeit inadequate empirical justification), as models for the calculus of experimentally verifiable propositions of quantum theory. It is seen that the quantum probability theory based on the logic of operations provides the proper mathematical framework for discussing the statistics of successive observations in quantum theory. We also indicate how a theory of quantum stochastic processes can be formulated in a way similar to the Kolmogorov formulation of the classical theory.
Elementary systems in general relativity
Authors: Maia M. D.
Published: March 2003

The problem of definition of elementary systems in general relativity is analyzed and some current theories are compared. A definition of dynamically free local elementary systems in general relativity is proposed, taking into account the influence of the gravitational field arising from the intrinsic dependence of the symmetry groups on the field.
Realizations of the central extension of the inhomogeneous symplectic algebra as time dependent invariance algebras of nonrelativistic quantum systems
Authors: Burdet G. , Perrin M.
Published: March 2003

We interpret the one-element central extension of the Lie algebra of the group of real inhomogeneous linear canonical transformations of phase space as a time dependent invariance algebra in the Schrödinger picture. By limiting to a two-dimensional space quantum system, we exhibit all the nonconjugate Hamiltonians contained in the symplectic algebra which, in the Heisenberg picture, plays the role of dynamical algebra and contains several known degeneracy and spectrum generating algebras.
Marginally singular integral equation with divided-difference kernel: A problem in N/D theory
Authors: Johnson Porter W. , Warnock Robert L.
Published: March 2003

We describe a method for solution of a linear integral equation of the form φ (s) =f (s)+(1/π) ℱ∞4{[B (s)−B (t)]/(s−t)} q (t) φ (t) dt, where q (t) = (1−4/t)1/2. The specified functions f (s) and B (s) have the asymptotic behaviors f (s) ∼f0 (lns)−α, α≳1, B (s) ∼b (lns)−1, s→∞; in addition, B is subject to smoothness conditions. The equation is analyzed on a Banach space S of continuous functions φ (s) which have asymptotes of the form φ0(lns)−α. It is found that the integral operator K is bounded but not compact on the space S, so that the equation is not of Fredholm type on S. We separate K into a noncompact part K1 and a compact part K2, and construct explicitly the inverse of 1−K1 by solving an associated differential equation. We then convert the original equation φ=f+Kφ into an equivalent Fredholm equation φ= (1−K1)−1(f+K2φ).
Second-quantization representation for a nonrelativistic system of composite particles. I. Generalized Tani transformation and its iterative evaluation
Authors: Girardeau M. D.
Published: March 2003

A method of constructing ’’atomic second quantization’’ representations is described, based on the introduction of redundant modes (’’ideal atom variables’’) which are then given physical content by carrying out a suitable unitary transformation, the ’’generalized Tani transformation.’’ In such a representation bound atoms or molecules are described by elementary Bose or Fermi operators, the field operators for nuclei and electrons referring only to unbound particles. The Hamiltonian thus obtained contains not only nucleus–nucleus, electron–electron, and nucleus–electron Coulomb interactions, but also atom–atom, atom–nucleus, and atom–electron Coulomb and exchange interactions, including breakup and recombination terms. All possible scattering and reaction channels are exhibited simultaneously in this transformed Hamiltonian. The method is applicable to any species or mixture of species of composite particles, but for simplicity the derivation is restricted here to the case of atomic hydrogen.
Borel summability and distribution
Authors: Hillion P.
Published: March 2003

This paper concerns the Borel summability of series expansions Σ∞0 fn ψn(x) =B f (x), where the polynomials ψn(x) defined on the interval I of the real line, are orthogonal in some space L2(I) and we look for conditions on fn such that f (x) is a distribution. As an application, a long standing ambiguity in the quantum theory of Coulomb scattering is solved.
Small energy denominators in interacting quantum systems: Bound states
Authors: Aks S. O/. , Varga B. B. , Sienicki J. J.
Published: March 2003

A perturbative method is developed for calculating bound states of interacting quantum systems, which is based on an analysis of terms with small energy denominators. An iterative scheme is formulated in a systematic manner which eliminates small energy denominators completely. The method is applied to the φ4 model of interacting bosons. The zeroth order solution of the equation of motion differs significantly from the usual free solution, and satisfies a different equation, the determining or bifurcation equation. The additional information contained in this zeroth approximation is used to calculate properties of bound states.
Nonmetrical dynamics. I
Authors: Pereira C. Martin
Published: March 2003

Einstein’s nonvacuum gravitational field equations are examined. The following nonmetrical existence theorem is proved. It is shown that given any analytic symmetric contravariant energy–momentum tensor density as a function of the space–time coordinates, a solution to the gravitational field equations always exists. Furthermore, this solution is such that the law of conservation of energy–momentum is satisfied. The new proof of the present paper makes no use of the coordinate transformation method used in a previous paper. The Cauchy–Kowalewsky existence theorem is used in the proof, and the literature on the Cauchy–Kowalewsky existence theorem is briefly reviewed. (Riquier’s existence theorem is avoided except for a brief discussion in the Appendix.) The initial value problem of Einstein’s equations is examined as part of the proof. The physical and mathematical meaning of this new proof is discussed. It is noted that the nonmetrical existence theorem disappears when Einstein’s equations are linearized whereas the familiar Lichnerowicz-type existence theorems survive linearization. This suggests that the nonmetrical existence theorem may add a new dimension to our understanding of the physical meaning of nonlinearity in Einstein’s equations.
Quantum theory of anharmonic oscillators. I. Energy levels of oscillators with positive quartic anharmonicity
Authors: Hioe F. T. , Montroll E. W.
Published: March 2003

This is an investigation of the energy levels of an anharmonic oscillator characterized by the potential (1/2) x2+λx4. Two regions of λ and n are distinguishable (n being the quantum number of the energy level) one in which the harmonic oscillator levels En=n+1/2 are only slightly distorted and the other in which the purely quartic oscillator form En?cλ1/3(n+1/2)4/3 (c being a constant) is only slightly distorted. Rapidly converging algorithms have been developed, using the Bargmann representation, from which energy levels in any (λ,n) (with λ≳0) regime can easily be computed. Simple formulas are also derived which give excellent approximations to the energy levels in various (λ,n) regimes.
Moment inequalities for ferromagnetic Gibbs distributions
Authors: Newman Charles M.
Published: March 2003

Moment inequalities analogous to Khintchine’s inequality (for sums of independent Bernoulli random variables) are obtained for a certain class of random variables which naturally arises in the context of ferromagnetic Ising models and φ4 Euclidean (quantum) field models in a positive external field. These results extend ones obtained previously which applied only to the mean zero (vanishing external field) case.
Null geodesic deviation. I. Conformally flat space–times
Authors: Peters P. C.
Published: March 2003

The equation of geodesic deviation is solved in conformally flat space–time in a covariant manner. The solution is given as an integral equation for general geodesics. The solution is then used to evaluate second derivatives of the world function and derivatives of the parallel propagator, which need to be known in order to find the Green’s function for wave equations in curved space–time. A method of null geodesic limits of two-point functions is discussed, and used to find the scalar Green’s function as an iterative series.
Propagation of high frequency surface waves along cylinders of general cross section
Authors: Morrison J. A.
Published: March 2003

The propagation of high frequency scalar surface waves along the generators of a homogeneous cylinder which has a cross-sectional boundary of nonconstant curvature is investigated. Asymptotic solutions are obtained to the reduced wave equation, subject to an impedance boundary condition at the surface of the cylinder. In the case of an open boundary curve for which the curvature attains its algebraic maximum at a single point, it is found that modes exist for which the disturbance is essentially confined to a region in the neighborhood of the point of maximum curvature, as well as to the neighborhood of the surface. The amplitude of the disturbance decays rapidly on either side of the point of maximum curvature, and the higher order modes have nulls. The case of closed boundary curves is also discussed. In a companion joint paper by L. O. Wilson and the author, the asymptotic procedures developed for the analysis of the scalar problem will be applied to the investigation of the propagation of elastic surface waves (Rayleigh waves) along a homogeneous isotropic cylinder with stress-free boundary. This problem arises in connection with guided acoustic surface waves.
Propagation of high frequency elastic surface waves along cylinders of general cross section
Authors: Wilson L. O. , Morrison J. A.
Published: March 2003

The propagation of high frequency elastic surface waves along the generators of a homogeneous isotropic cylinder which has a cross-sectional boundary of nonconstant curvature is investigated. The boundary surface is stress-free and the surface waves, or Rayleigh waves, are disturbances whose amplitudes decay rapidly with depth into the cylinder. In the case of an open boundary curve for which the curvature attains its algebraic maximum at a single point, it is found that modes exist for which the disturbance is essentially confined to a region in the neighborhood of the point of maximum curvature, as well as to the neighborhood of the surface. The amplitude of the disturbance decays rapidly on either side of the point of maximum curvature. The application of these high frequency asymptotic results to the case of a closed boundary curve is discussed. Particular cases will be investigated in more detail in a subsequent paper.
Dilatations and the Poincaré group
Authors: Boya Luis J. , Cariñena Jose F. , Santander Mariano
Published: March 2003

We discuss the projective unitary representations of the Weyl group (Poincaré group enlarged with dilatations).
Properties of linear representations with a highest weight for the semisimple Lie algebras
Authors: Gruber B. , Klimyk A. U.
Published: March 2003

A theory for the representations with a highest and/or lowest weight is given for the semisimple complex Lie algebras (and their real forms). These representations are either irreducible finite-dimensional, irreducible infinite-dimensional or reducible, but not completely reducible, infinite-dimensional (called elementary representations), depending upon the property of the associated highest (or lowest) weight Λ. No restriction is made to those representations of the semisimple Lie algebras which can be integrated to form representations of the corresponding Lie group. The algebra A1 is chosen (Sec. III) as a simple and familiar example upon which, however, much of the proof for the results obtained for the theory of representations with a highest (and/or lowest) weight for the general case of a semisimple Lie algebra rests (Sec. IV). It is demonstrated that the irreducible representations D (Λ) with a highest (and/or lowest) weight Λ of the semisimple Lie algebras decompose with respect to any (regularly) embedded subalgebra of the type A1 in the manner that either (a) the subrepresentations subduced on A1 are all irreducible finite-dimensional, or (b) all infinite-dimensional. If for case (b) the complex number Mα ≡ 2(M,α)/(α,α), α the (simple) root of A1 and M a weight of D (Λ) extremal with respect to A1, is not a nonnegative integer, then the representation subduced on A1 is irreducible. If, however, Mα is a nonnegative integer, then a reducible but not completely reducible, representation is subduced on A1. Based upon the results of Sec. IV a generalization of Freudenthal’s formula is obtained in Sec. V, valid for irreducible infinite-dimensional representations with highest (or lowest) weight. In Sec. VI generalizations are given of Racah’s recurrence relation for the multiplicity of weights, Weyl’s character formula and Kostant’s formula for the multiplicity of weights for infinite-dimensional irreducible representations with a highest (or lowest) weight of the semisimple Lie algebras. These formulas are derived utilizing theorems and lemmas obtained by Verma, I. M. Gel’fand, S. I. Gel’fand, Bernstein, Harish-Chandra and the results of Sec. IV. In Sec. VII some of the infinite-dimensional representations of the algebra A2 are discussed as examples, employing the geometrical methods developed by Antoine and Speiser and by Biedenharn and others.
Path integral method for superfields
Authors: Huq M.
Published: March 2003

A path integral method for superfields has been developed. This is made possible using a suitable definition for functional differentiations in the 8-dimensional space of x and ϑ. The Ward–Takahashi identities arising from the invariance under supersymmetry transformations are shown to take very neat and compact forms for the Green’s functions of the superfields.
Hamiltonian formulation of the classical two-charge problem in straight-line approximation
Authors: Kennedy Frederick James
Published: March 2003

The Newtonian equations of motion expressing the interaction of electric charges correct up to terms of order e2, the product of the charges, is cast into Hamiltonian form. Lorentz transformations are canonically represented to order e2, but as anticipated by the zero-interaction theorem, there is no canonical transformation from the canonical variables to the charges’ physical positions.
On properties of the singularity of the ground state in certain classical Heisenberg models
Authors: Wreszinski Walter F.
Published: March 2003

We prove that for both the classical ferro- and antiferromagnetic Heisenberg models the infinite volume limit of the ground state energy per unit volume of the system (Hamiltonian plus λ times an operator) is not differentiable at zero in λ for some operators. This characterization of the singularity at T=0, which corresponds to Fisher’s for positive temperature, adds to a number of others, which are to some extent analogous to the several characterizations of phase transitions at T≳0. A comment is made upon a related open problem concerning the ground state of the quantum antiferromagnetic Heisenberg chain.
A toroidal solution of the vacuum Einstein field equations
Authors: Thorne Kip S.
Published: March 2003

This paper presents a new solution to the vacuum Einstein field equations for the static external gravitational field of a toroidal singularity. This solution is unique among known toroidal solutions in that the singularity is locally cylindrically symmetric; near it the spacetime geometry becomes that of an infinite line mass (’’Levi-Civita metric’’).
Canonical transformations and accidental degeneracy. IV. Problems with continuous spectra
Authors: Moshinsky M. , Patera J.
Published: March 2003

In the present series of papers we have been trying to bring out the unifying role of groups of canonical transformations in the understanding of problems of accidental degeneracy in quantum mechanics. In Paper III of this series we achieved our purposes for two-dimensional problems with discrete spectra. In the present paper we turn our attention to problems with continuous spectra. There is the well-known case of the free particle in the full plane in which the accidental degeneracy is due to the Euclidean group in two dimensions, E (2). We show that in this problem the accidental degeneracy can also be explained by an O (2,1) group of canonical transformations which provides a clue of the approach to more general problems. We also derive explicitly the group O (2,1), and not only its Lie algebra, associated with the accidental degeneracy of the Coulomb problem in two dimensions. The procedure followed in the above problems is ’’sui generis’’ and does not provide a general approach. For the latter we discuss two new problems with continuous spectra that have accidental degeneracy: the free particle in a sector of angle π/q, q integer, of the plane and the Calogero problem with continuous spectrum. For both of these problems we find the canonical transformations that map them on the free particle in the full plane. It turns out that their accidental degeneracy is explained then by the O (2,1) group of the latter problem, that we mentioned above, rather than by E (2). The procedures developed seem general enough to encompass other problems of accidental degeneracy in configuration spaces of two or more dimensions.
On the solution of the phase retrieval problem
Authors: Hoenders B. J.
Published: March 2003

It is shown that the intensity in the image plane of a microscope determines uniquely the phase of the corresponding image wavefunction up to an over-all phase. This result is obtained using the a priori information that both the image wavefunction and the unperturbed wavefunction in the Fraunhofer plane are band-limited and that we have some a priori knowledge about the intensity at the rim of the diaphragm in the Fraunhofer plane. If we have no useful a priori information about the wavefunction in the Fraunhofer plane, unique phase reconstruction is possible from two exposures, corresponding to two different values of the defocusing.
Remarks on the Green’s functions for the cubic lattices
Authors: Mano Koichi
Published: March 2003

A simpler way of finding the Green’s function G (2p,0,0) for the body-centered and face-centered cubic lattices as the square and product, respectively, of the pth derivatives of two complete elliptic integrals of the first kind is pointed out. The key relations required for these results are the Clausen’s and Brafman’s formulas for the body-centered and face-centered cubic lattices, respectively.
On Ursell’s combinatorial problem
Authors: MacGillivray A. D.
Published: March 2003

A combinatorial problem considered by H. D. Ursell in his seminal paper on cluster theory [Proc. Camb. Phil. Soc. 23, 685 (1927)] is studied. Ursell’s analysis, which is not rigorous, is described by Fowler and Guggenheim as being far from simple. In this paper we arrive at Ursell’s result using a method which is straightforward, yet completely rigorous.
Killing vectors in empty space algebraically special metrics. II
Authors: Held A.
Published: March 2003

Empty space algebraically special metrics possessing an expanding degenerate principal null vector and Killing vectors are investigated. Attention is centered on that class of Killing vector (called nonpreferred) which is necessarily spacelike in the asymptotic region. A detailed analysis of the relationship between the Petrov–Penrose classification and these Killing vectors is carried out.
Weyl conform tensor of δ=2 Tomimatsu–Sato spinning mass gravitational field
Authors: Economou John E. , Ernst Frederick J.
Published: March 2003

Extremely simple expressions are presented for the hitherto uncalculated invariants associated with the Weyl conform tensor of the δ=2 Tomimatsu–Sato solution of Einstein’s field equations.
Black holes in a magnetic universe
Authors: Ernst Frederick J.
Published: March 2003

We present a general procedure for transforming asymptotically flat axially symmetric solutions of the Einstein–Maxwell equations into solutions resembling Melvin’s magnetic universe. Specific applications yield metrics associated with black holes in a magnetic universe. It is hoped that these solutions will be of interest to astrophysicists studying gravitational collapse in the presence of strong magnetic fields.
The spectrum of the Liouville–von Neumann operator
Authors: Spohn Herbert
Published: March 2003

We relate the pure point spectrum, the singularly continuous, and the absolutely continuous part of the spectrum of the Liouville–von Neumann operator [H,⋅] to the respective parts of the spectrum of the Hamiltonian operator H. As a consequence of this result we obtain a theorem about the weak* limit of the time evolution W (t) of a normal state W for t → ∞.
Generalized Källén–Pauli equation
Authors: Raju Cvavb Chandra
Published: March 2003

The Källén–Pauli (KP) equation in the 2V particle model was solved. In addition to satisfying all the requirements of the integral equation, this solution is found to be reducing to that of the ordinary KP equation. Unlike earlier authors, we found that there is a resonance in the Vϑ sector also. The solution given here shows that the deductive method used in the case of the ordinary KP equation does not hold good in the present case. The uniqueness of the solution is yet to be proved.
Goldstone’s theorem for quantum spin systems of finite range
Authors: Wreszinski Walter F.
Published: March 2003

Goldstone’s theorem is proved for quantum spin systems of finite range with rotationally invariant Hamiltonian under the assumptions that the ground state of the infinite system is unique, is invariant under a subset of the translation group (which depends on the system), and has long range order.
An application of the method of geometrical optics to the scattering of plane electromagnetic waves off cylindrically confined cold plasmas
Authors: Kriegsmann Gregory A.
Published: March 2003

In this paper we apply the method of geometrical optics to study the scattering of plane electromagnetic waves off a cylindrically confined cold plasma. For simplicity, we assume two types of incident polarization. In both cases scalar second order elliptic partial differential equations describe the fields. These problems are studied in the asymptotic limit aω/c→∞ (where a is the radius of the cylinder, ω is the frequency of the incident plane wave, and c is the velocity of light in free space). We furthermore assume a quadratic plasma density. This allows us to calculate explicitly the rays, amplitudes, caustics, and other features which arise in the geometrical optics approach. The assumed density also gives rise to the interesting cutoff and resonance phenomenon. Thus the amplification of the electromagnetic fields is observed. This phenomenon may have applications in laser fusion.
On analytic nonlocal potentials. IV. A closed contour dispersion relation
Authors: Yao Te Hai
Published: March 2003

We give a class of analytic nonlocal potentials which give rise to a closed contour s-wave dispersion relation for sufficiently small coupling constant.
On the positive energy conjecture
Authors: Jang Pong Soo
Published: March 2003

A theorem, to the effect that the positive energy conjecture is true for any space–time possessing a flat spatial hypersurface, is obtained.
Properties of massless relativistic fields under the conformal group
Authors: Post G.
Published: March 2003

Using the ladder representations of SU(2,2), we derive explicit transformation laws for massless free fields with arbitrary helicities under global conformal transformations.
An upper bound on the energy gap in the (λφ4+σφ2)2 model
Authors: Klein Abel , Landau Lawrence J.
Published: March 2003

Let ΔE (λ,σ,m0) be the energy gap in the infinite volume quantum field theory with bare mass m0 and interaction λφ4+σφ2 in two space–time dimensions, obtained with either full-Dirichlet or half-Dirichlet boundary conditions. Then ΔE (λ,σ,m0)/m0<exp[σ/(m20 +3λ/Π)] for all λ≳0 and σ real. In particular ΔE (λ,0,m0) <m0 for all λ≳0 and ΔE (λ,σ,m0)/m0→0 as σ→−∞. For half-Dirichlet boundary conditions one also has ΔE (λ,σ,m0)/m0< (1+2σ/m20)1/2 for σ⩽0,λ≳0. In each pure phase of a (λφ4)2 theory, let mphys be the energy gap and 〈φ〉 the expectation of the field. Then mphys/m0<exp[2Π〈φ〉2/(1+Πm20/3λ)].
Bogoliubov inequality for unbounded operators and the Bose gas
Authors: Bouziane M. , Martin Ph. A.
Published: March 2003

We provide the mathematical arguments which are needed to obtain a rigorous proof of the absence of condensation in a one- and two-dimensional Bose gas of particles having superstable interactions.
Expansion technique for crystal surface problems
Authors: Trullinger S. E.
Published: March 2003

An expansion technique for representing functions localized at crystal surfaces is developed by making use of the Gottlieb polynomials to construct a basis set of orthonormal functions of a discrete variable (the integers from zero to infinity). Using these functions, it is shown that a large class of surface problems can be recast in the form of an eigenvalue problem involving a matrix whose elements can be obtained in closed form. The matrix size can be minimized by varying a parameter upon which the basis functions depend. As an example, the dispersion curve of a ferromagnetic spin wave localized at the (001) surface of an fcc crystal is calculated.
Lie algebras in the Schrödinger picture and radial matrix elements
Authors: Chacón E. , Levi D. , Moshinsky M.
Published: March 2003

The main objective of this paper is to derive from a unified viewpoint particular one- and two-body radial matrix elements with respect to oscillator and Coulomb states. All the results have been obtained previosly using either generating functions of Laguerre polynomials or group theoretical methods related to particular realizations of the Lie algebras associated with those states. We show, though, in this paper how some of the realizations proposed can be derived from physical considerations. The main idea is to translate well-known realizations, in the Heisenberg picture, to the corresponding ones in the Schrödinger picture. The latter realizations allow us to define indecomposable (i.e., not completely reducible) and irreducible tensors of the Sp(2) [or equivalently the SU(1,1)] group for the one- and two-body problem respectively. The evaluations of the radial matrix elements becomes then just a matter of applying the Wigner–Eckart theorem, giving rise to Wigner coefficients of SU(1,1) that have been discussed extensively in the literature.
Formal solutions of inverse scattering problems. II
Authors: Prosser Reese T.
Published: March 2003

The formal solutions of inverse scattering problems presented in Paper I [J. Math. Phys. 10, 1819 (1969)] are shown here to converge in certain cases of potential scattering for sufficiently weak potentials, and in certain cases of refractive scattering for sufficiently weak variations in the index of refraction. The solutions for the cases of boundary scattering, on the other hand, are not likely to converge, because there is no way to make the effect of the boundary sufficiently weak.