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Canonical transforms. IV. Hyperbolic transforms: Continuous series of SL(2,R) representations
Authors: Wolf Kurt Bernardo
Published: March 2003

We consider the sl(2,R) Lie algebra of second-order differential operators given by the Schrödinger Hamiltonians of the harmonic, repulsive, and free particle, all with a strong centripedal core placing them in the Cεq continuous series of representations. The corresponding SL(2,R) Lie group is shown to be a group of integral transforms acting on a (two-component) space of square-integrable functions, with an integral (matrix) kernel involving Hankel and Macdonald functions. The subgroup bases for irreducible representations consist of Whittaker, power, Hankel, and Macdonald functions. We construct the operator which intertwines this realization of SL(2,R) with the more familiar Bargmann realization on functions on the unit circle. This operator implements the canonical transformation of the above Schrödinger systems to action and angle variables.
Ewald evaluation and Poisson summation formulas for a class of near-Coulombic lattice sums
Authors: Hall George L.
Published: March 2003

The class of lattice sums studied is defined by Fn,τ=∑∞τ′[(1/τn) −(1/Ω) ∫τ(d3r/rn)] −(1/Ω) ∫0(d3r/rn) for 1⩽n⩽2, where {τ⃗} denotes a three-dimensional Bravais lattice with volume Ω per lattice point, {γ⃗} denotes its reciprocal lattice normalized with exp(iτ⃗⋅γ⃗) =1 with volume v= (2π)3/Ω per lattice point, and except for the case n=1 the cellular integrations are taken over centered cells, proximity or primitive. Ewald evaluation is discussed for all cases and reciprocal relations between the Fn,τ and the F(3−n),γ are supplied for all n. These relations are essentially Poisson summation formulas (PSF) with modifications for the cases n=1,2. The method of securing the PSF’s involves a direct, and an inverse, use of an Ewald method coupled with careful analysis of all interchanges of integrations and summations. The method should be useful for similar studies on other complex sums which, like these, are not very amenable to direct application of a PSF with assurance of validity.
On Mathieu equation with damping
Authors: Hsieh D. Y.
Published: March 2003

A direct variational method is applied to the linear and nonlinear Mathieu equation with damping. It is found that the nature of the periodic solutions and the characteristic curves are modified due to the presence of the damping. A threshold value of β is required to overcome the damping for the existence of the periodic solutions. Stability analyses for the periodic solutions are also carried out.
Inverse problem for the reduced wave equation with fixed incident field
Authors: Weston V. H.
Published: March 2003

The inverse problem for the reduced wave equation Δu+k2n2(x) u=0 is examined when n (x) is assumed to be real continuous and equal to unity outside some prescribed compact domain D0 in R3. The case where a finite set of measurements of the scattered or total field (generated by a single incident wave at a fixed frequency) at points outside D0 is considered. It is shown that the inverse problem can be expressed in terms of a system of linear functional equations, plus a quadratic nonlinear integral equation. By imposing an additional criterion for uniqueness of the functional equations and placing certain restrictions on the size of the measured data, a solution of the system can be generated by successive approximations. It is shown that the iterative scheme yields correction terms to the solution that are obtained by the common linearized approximation where n (x) is assumed to be a very small perturbation of a known quantity and only linear terms are retained.
Subsystems in physics described by bilinear maps between the corresponding vector spaces
Authors: Aerts Dirk
Published: March 2003

We show that whenever a physical system is composed of two subsystems, there exists a (σ1,σ2) -linear map between their generalized Hilbert spaces which describes this composition. As a consequence, subsystems of a physical system described by a generalized Hilbert space over a division ring K are always described by a generalized Hilbert space over a subdivision ring of K.
A probabilistic formulation of quantum theory
Authors: Bach Alexander
Published: March 2003

By means of integration in the Hilbert space H which is associated to a quantum mechanical system we demonstrate that the expectation value of quantum mechanics, EW(A) =tr(WA), can be expressed as EW(A) =ℱH dμW(φ)   (1/2) 〈φ‖A‖φ〉 Here μW denotes a probability measure in H which is determined by the statistical operator. We discuss this hidden variables representation of quantum mechanics.
Positive quantum joint distributions
Authors: Cohen Leon , Zaparovanny Y. I.
Published: March 2003

We demonstrate the existence of positive phase space density functions which yield the quantum mechanical marginal distributions of position and momentum.
A Lie superalgebraic interpretation of the para-Bose statistics
Authors: Ganchev A. Ch. , Palev T. D.
Published: March 2003

We show that n pairs of para-Bose operators generate the classical simple orthosymplectic Lie superalgebra osp(1,2n) ≡B (0,n). The creation and annihilation operators are negative and positive root vectors, respectively, and they span a basis in the odd part of B (o,n).
On the extremum problem of the energy functional on the set of Slater determinants
Authors: Fonte Giacomo
Published: March 2003

Differential properties of the energy functional on the set of Slater determinants are defined according to a new differential calculus on complex Banach spaces previously considered. Such properties permit us to derive, with a new and rigorous procedure, the Hartree–Fock equations and to define an iterative method of the gradient type to minimize this functional on the above set. For a large class of one-body and two-body potentials, including Coulombic potentials, we show that our gradient method yields a decreasing sequence of values of the energy functional and a sequence of approximate solutions of the Hartree–Fock equations. The convergence of the latter sequence to a solution of these equations is proved in any finite-dimensional subspace.
Generalized Grassmann algebras with applications to Fermi systems
Authors: Ohnuki Y. , Kamefuchi S.
Published: March 2003

Generalized Grassmann numbers xi (i=1,2,⋅⋅⋅,n) are defined as those satisfying the relations xixj=ηijxjxi with ηij =−(+ or −) for i=j(i≠j). Ordinary Grassmann numbers correspond to a special case ηij=− for all i, j. Mathematical properties of such numbers are discussed in detail, and it is found that most of the results known for the ordinary case can naturally be extended to the general case. Applications are made to a description of general Fermi systems where the commutation relations belong to an arbitrary anomalous case.
Para-Grassmann algebras with applications to para-Fermi systems
Authors: Ohnuki Y. , Kamefuchi S.
Published: March 2003

Grassmann algebras of the usual kind are generalized to what are to be called para-Grassmann algebras. Para-Grassmann numbers are defined as those satisfying the trilinear commutation relations that resemble the para-Fermi commutation relations. Basic mathematical properties of these algebras are studied in detail on the basis of generalized Grassmann algebras discussed in the preceding paper. Applications are made to description of para-Fermi systems, and it is found that such systems can completely be described in what we call the para-Grassmann representation.
A Galileian formulation of spin. II. Explicit realizations
Authors: Brooke J. A.
Published: March 2003

This paper is the follow-up of an earlier one. It reviews some of the lesser known ideas involved in the representation theory of Clifford algebras and applies these ideas in computing explicit realizations of the spin groups and the Lie algebras appearing in the previous paper.
A note on the symmetries of the 3j and 6j coefficients. I
Authors: Venkatesh K.
Published: March 2003

It is shown that the study of the symmetries of the 3j coefficient in terms of the set of six 3F2(1)’s derived by us introduces a six-to-one homomorphism of the 72-element group of symmetries of the 3j coefficient on to the 12 permutations of parameters of a single 3F2(1) series of the set. Also, the study of the symmetries of the 6j coefficient in terms of the set of three (four) 4F3(1)’s derived by us introduces a three (four) -to-one homomorphism of the 144-element group of symmetries of the 6j coefficient on to the 48(36) allowed permutations of parameters of a single 4F3(1) series of the set.
Permutational symmetry of many particle states
Authors: Sarma C. R. , Sahasrabudhe G. G.
Published: March 2003

The symmetry of the Nth rank tensor basis for an irreducible representation of U (n) under the operations of the permutation group has been investigated. It has been found that symmetrized linear combinations of the elements of the matrix algebra of SN lead to a tensor basis for U (n) yielding the same matrix elements as the Gel’fand–Tsetlin basis for the generators Ei,i±1 of U (n). Based on these developments an algorithm has been developed for directly determining the matrix elements of the generators Eij(j≠i±1) of U (n) using a pattern calculus.
On the dynamical symmetries of the Kepler problem
Authors: Akyildiz Yilmaz
Published: March 2003

We try to understand the geometry of the SO(n+1,2) action on the Kepler Manifold of the n-dimensional hydrogen atom. We show that the SO(n+1,2) symmetry of the Kepler Problem is closely related to the fact that the geodesic flow on T*Sn is periodic. We also exhibit the orbit picture analog of the peculiar property of the corresponding SO(n+1,2) representation; that is, it stays irreducible when restricted to SO(n+1,1) subgroups.
Lorentz deformation in the O(4) and light-cone coordinate systems
Authors: Kim Y. S. , Noz Marilyn E. , Oh S. H.
Published: March 2003

A formalism is developed for describing Lorentz deformation properties of extended hadrons in terms of solutions of the harmonic oscillator equation in the O(4) and light-cone coordinate systems. The physical hadronic wave function discussed in previous papers is written as a linear expansion of orthonormal functions in those coordinates which form representations of compact groups. The separability of the oscillator equation is shown to play the essential role in developing the proposed mathematics.
The Baker–Campbell–Haussdorff formula for the SU(2) supergroups
Authors: Aragone C. , Chela-Flores J. , Restuccia A.
Published: March 2003

In the theory of supersymmetric SU(2) Yang–Mills fields described on the 8th dimensional superspace, the local gauge transformations constitute a group whose Lie algebra has its coefficients in the Weyl-spinorial-Grassmann algebra. The coefficients can be taken to be chiral elements, real or complex. This last case is the most general one and contains all others of interest for developing physical models. We present here a Baker–Campbell–Haussdorff formula for the complex SU(2) supergroup. The formula gives the finite form for each element of the supergroup in terms of the local fields entering into the infinitesimal (complex) superscalar generator. Thereafter we exhibit, as a particular case, a BCH-expression for the real SU(2) supergroup which completes previous results already reported for the chiral SU(2) supergroups. Moreover, it is shown that, despite the appearance of apparently nonperiodic polynomial terms, the global variety of the real SU(2) supergroup is contained in a variety having the same compact bosonic width as the usual SU(2) pure bosonic variety.
On the scattering of plane electromagnetic waves off cylindrically confined cold plasmas with overdense and steep densities
Authors: Kriegsmann Gregory A.
Published: March 2003

In this paper we study the scattering of plane electromagnetic waves off a cylindrically confined cold plasma. The plasma density is taken to be overdense and very steep. This causes the cutoff radius, r0, to be within a fraction of a wavelength from the cylindrical boundary. 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→∞ with 0<1−r0=O (c/aω). Here a is the radius of the cylinder, ω is the frequency of the incident radiation, and c is the velocity of light in free space. We develop an asymptotic technique which reduces the partial equations to ordinary differential equations within the plasma. Our method is a blend of geometrical optics and boundary layer techniques. Outside the plasma we use straight geometrical optics to describe the scattered field.
Bilateral classes
Authors: Hässelbarth W. , Ruch E. , Klein D. J. , Seligman T. H.
Published: March 2003

We introduce ’’bilateral classes’’, a new concept for classifying the elements of groups. Bilateral classes are orbits in a group G under the action of any given subgroup of the direct product G×G. The classification concept presented here encompasses conjugacy classes, cosets, double cosets, and Ree’s σ-classes as particular cases. It has an interpretation as a classification scheme for bijections between G-sets under the aspect of symmetry equivalence due to symmetries in both the domain and the range. While double cosets and conjugacy classes correspond to the case of no or complete correlation between operations of the two symmetry groups, our concept also covers the general case of partial correlation. The scope of generalization corresponds to applications in physics. Expressions for the number of bilateral classes are given.
Commutation properties of 2-parameter groups of isometries
Authors: Ernotte P.
Published: March 2003

We prove a stronger version of a result due to B. Carter: A cyclic subaction must commute with any 1-parameter subgroup of a 2-parameter group of isometries.
Clebsch–Gordan coefficients for corepresentations. I⊗II
Authors: Dirl R.
Published: March 2003

A general method is given to determine quite generally Clebsch–Gordan coefficients for corepresentations in terms of such ones of the normal subgroup, where the considered Kronecker products are composed of corepresentations of type I and II.
Clebsch–Gordan coefficients for corepresentations. III⊗III
Authors: Dirl R.
Published: March 2003

By means of a general method, Clebsch–Gordan coefficients for corepresentations are traced back by simple unitary transformations to convenient Clebsch–Gordan coefficients for the normal subgroup. The considered Kronecker products are composed of corepresentations of type III only.
Systematic improvement of Hall–Post–Stenschke lower bounds to eigenvalues in the few-body problem
Authors: Hill Robert Nyden
Published: March 2003

A method for systematically improving Hall–Post–Stenschke lower bounds to the bound state eigenvalues of three-body Schrödinger equations is given. The improved bounds are obtained by solving coupled one variable integral equations; the bounds get better as the number of coupled equations is increased. The method generates explicit wave functions which can be used to obtain complementary upper bounds via the Rayleigh–Ritz variational method. Either identical or nonidentical particles can be handled. The method is illustrated by calculations for three identical particles bound by Hooke’s law forces. A brief discussion of extensions to more than three particles is given.
Planar limit for SU(N) symmetric quantum dynamical systems
Authors: Marchesini G. , Onofri E.
Published: March 2003

The spectrum of a quantum mechanical Hamiltonian with SU(N) symmetry is studied in the limit N→∞. A complete description is given for the states which transform according to (i) the scalar representation (singlet states) and (ii) the adjoint representation of SU(N) (’’adjoint’’ states). The eigenvalues of the singlet states are equally spaced with a finite gap ω (g) as N→∞. The spectrum of the adjoint states is equally spaced asymptotically for large excitations with the same gap ω (g). The first excitation of the adjoint states is lower than the first singlet excitation. An accidental degeneracy appears which is removed by 1/N corrections. We compute explicitly this splitting for the singlet states. In this formulation 1/N2 plays a role similar to h/ so that all computed quantities are related to the corresponding classical system. All the quantities which enter in the calculation of the spectrum are analytic near g=0 with the same radius of convergence.
Aesthetic field theory; the problem of spatial inversions
Authors: Muraskin M.
Published: March 2003

In a previous paper we studied a set of origin point data that was invariant under three-dimensional rotations. This data, however, was not, in general, invariant under spatial inversions. We can obtain another set of data from the original data by performing a spatial inversion. Both sets of data can be incorporated into aesthetic field theory without compromising the basic ideas of the theory, by introducing complex fields. It is found that this combined set of data leads to computer solutions only slightly more complex than the corresponding real case.
Spherically symmetric space–times with vanishing curvature scalar
Authors: Goenner Hubert , Havas Peter
Published: March 2003

A systematic study of spherically symmetric space–times with vanishing curvature scalar R is undertaken. The solutions of the equation R=0 are classified according to the number of different eigenvalues of the Ricci tensor Rαβ (which for spherical symmetry necessarily possesses one double eigenvalue) as well as to the vanishing or nonvanishing of the conformal curvature tensor. The case of conformally flat spherically symmetric solutions is fully integrated, as well as the case of one quadruple or two double eigenvalues. For the case of one triple and one single eigenvalue various classes of solutions are obtained, and for that of one double and two single eigenvalues a number of particular solutions with two free functions of one variable are found. It is noted that the methods developed in this paper allow the full integration of the case of plane symmetric solutions of R=0.
Hierarchical equations of evolution of an anharmonic system
Authors: Esposito R. , Pulvirenti M.
Published: March 2003

We investigate the evolution of the states of a system of infinitely many anharmonic oscillators via a hierarchy of equations similar to the BBGKY one. We prove an existence theorem for the solutions in the L1 sense for a large class of initial data.
An application of the maximum principle to the study of essential self-adjointness of Dirac operators. II
Authors: Landgren J. J. , Rejto P. A. , Klaus Martin
Published: March 2003

In the present part II of this paper we prove the remaining part of the essential self-adjointness theorem stated in part I. This proof makes essential use of the maximum principle.
On the remarkable nonlinear diffusion equation (∂/∂x)[a (u+b)−2(∂u/∂x)]−(∂u/∂t)=0
Authors: Bluman George , Kumei Sukeyuki
Published: March 2003

We study the invariance properties (in the sense of Lie–Bäcklund groups) of the nonlinear diffusion equation (∂/∂x)[C (u)(∂u/∂x)]−(∂u/∂t) =0. We show that an infinite number of one-parameter Lie–Bäcklund groups are admitted if and only if the conductivity C (u) =a (u+b)−2. In this special case a one-to-one transformation maps such an equation into the linear diffusion equation with constant conductivity, (∂2ū/∂x̄2)−(∂ū/∂t̄) =0. We show some interesting properties of this mapping for the solution of boundary value problems.
On choosing the points in product integration
Authors: Sloan Ian H.
Published: March 2003

A product-integration rule for the integral ℱbak (t)  f (t)  dt is a rule of the form 𝒥ni=1wi f (ti), with the weights w1,...,wn chosen so that the rule is exact if f is any linear combination of a chosen set of functions φ1,...,φn . For some choices of {φj}, including the polynomial case, the points {ti} need to be carefully chosen if reliable results are to be obtained. In this paper known convergence results for the polynomial case with well-chosen points are summarized and illustrated, and extended to some nonpolynomial cases, including one proposed by Y.E. Kim for use in solving the three-body Faddeev equations. The convergence theorems yield practical prescriptions for choosing the points {ti} .
Decaying states in the rigged Hilbert space formulation of quantum mechanics
Authors: Bohm A.
Published: March 2003

Within the rigged Hilbert space formulation of quantum mechanics idealized resonances (without background) are described by generalized eigenvectors of an essentially self-adjoint Hamiltonian with complex eigenvalue and a Breit–Wigner energy distribution. This establishes the link between the S matrix description of resonances by a pole and the usual description of states by vectors, overcomes theoretical problems connected with the deviation from exponential law and simplifies the calculation of the decay rate formula.
Resonances in the Klein–Gordon theory of the relativistic Stark effect
Authors: Graffi S. , Grecchi V. , Maioli M.
Published: March 2003

Existence of resonances is proved for the time-independent Klein–Gordon equation describing the interaction of a charged particle with an external uniform field of small strength F in addition to the Coulomb attraction. It is further shown that the resonances reduce to the exactly known bound states of the problem as F→0, and to the resonances of the nonrelativistic Stark effect as c→∞.
Continuum calculus. IV. The Laplace transform method in the evaluation of the Feynman path integrals with a Gaussian measure and applications in quantum mechanics
Authors: Lee L. L.
Published: March 2003

The continuum calculus, proposed earlier [J. Math. Phys. 17, 1988 (1976)], is here applied to the development of a functional version of Laplace transform and a method for the evaluation of path integrals containing a Gaussian-like measure. Two methods in functional integration are proposed. First, the Gaussian integral for polynomial functionals and consequently functionals that can be expanded in Taylor series are examined. Formula (2.12) is derived. Next, we define the Laplace transform in the function space through the weak distribution formulation of Skorohod. Comparison of both approaches enables us to determine the expression of the functional integeral through a series of Laplace transformations. The second formula is given in Eq. (4.5). The latter formula is applicable to all Laplace transformable functionals. As an illustration of the utility of the formulas derived, we evaluate the integral of a cosine functional by methods 1 and 2, and obtain consistent results. Further applications to quantum mechanics are also presented. We examine the cases of a free particle,the quantum harmonic oscillator, the forced oscillator, and charged particle in a magnetic field. In all cases, we obtain correct results in comparison with known expressions. A numerical procedure is employed in the calculation of infinite products. The usefulness of the p-integral method is stressed.
An exactly solvable one-dimensional three-body problem with hard cores
Authors: Hill Robert Nyden
Published: March 2003

Three identical particles in one dimension interact via a potential which is infinite whenever one or more of the interparticle separations is less than a or greater than b, and zero when all interparticle separations lie between a and b. Their Schrödinger equation is solved by reducing it to the exactly solvable problem of the two dimensional Helmholtz equation inside an equilateral triangle.
Test of a method for finding lower bounds to eigenvalues of the three-body problem
Authors: Bernard James E. , Hill Robert Nyden , Barnhill III Maurice V.
Published: March 2003

The simplest version of a method for systematically improving Hall–Post–Stenschke (HPS) lower bounds to eigenvalues is tested on an exactly soluble one dimensional three-body problem with hard cores. Significant improvement over the HPS bound is obtained, but considerable room for additional improvement remains.
Gauge theory in Hamiltonian classical mechanics: The long-range fields
Authors: Boisseau B.
Published: March 2003

The generalization of a previous work leads us to a gauge theory in Hamiltonian classical mechanics whose elements are: First, the infinitesimal canonical transformations considered as gauge transformations and, second, an infinite sequence of gauge potentials. A hierarchy of approximation orders allows the physical interpretation of this formalism. At the zero order we obtain the electromagnetic field, at the first order the electromagnetic and gravitational fields. At the second order a new field is added to the former ones. We have studied the new physical features involved by this hypothetical field, in the motion of a classical particle. In the approximation of a Keplerian motion, we have given the period. This last result could be eventually used to test this theory.
Coherent states and projective representation of the linear canonical transformations
Authors: Daubechies Ingrid
Published: March 2003

Using a family of coherent state representations we obtain in a natural and coordinate-independent way an explicit realization of a projective unitary representation of the symplectic group. Dequantization of these operators gives us the corresponding classical functions.
Prolongation structure for a nonlinear equation with explicit space dependence
Authors: Chowdhury A. R. , Roy T.
Published: March 2003

A nonlinear Schroedinger equation with a term depending explicitly on a space variable, e.g., iψt+ψxx+(−2αx+2‖E‖2) ψ=0 with ψ=Eeiφ has been treated in the language of differential forms and prolongation. The inverse scattering equations previously invented by Liu and Chen are obtained. A unique feature of the analysis is explicit space–time dependence of the pfaffian forms.
Matching of the plane symmetric static and homogeneous vacuum solutions of the Einstein field equations
Authors: Carlson George T. , Safko John L.
Published: March 2003

The Taub plane symmetric static and homogeneous vacuum solutions are matched on a natural hypersurface. The space–times obtained in this way have distribution valued curvature tensors along the joining hypersurfaces. Our treatment of this problem follows Taub’s presentation of space–times with distribution valued curvature tensors. We find that the surfaces of the join may be interpreted as thin null pressureless fluid shocks. The nature of these surfaces are further investigated by examining the behavior of geodesics crossing the surfaces.
Einstein–Cartan theory in the spin coefficient formalism
Authors: Gambini R. , Herrera L.
Published: March 2003

The field equations of the Einstein–Cartan theory are written down using the spin coefficient formalism developed by Newman and Penrose for the Einstein theory. The irreducible spinor decomposition of the Riemann tensor in a U4 space is obtained.
A functional integral formalism for quantum spin systems
Authors: Dacol Dalcio Kisling
Published: March 2003

A functional integral algorithm for the partition function and for the temperature Green’s function generating functional of spin 1/2 systems is developed. Starting from the representation of the spin operators in terms of fermion creation and destruction operators we write the above mentioned objects as functional integrals over an infinite dimensional Grassmann algebra. We show how to eliminate the integration over the Grassmann algebra in favor of an integration over function space. The use of the formalism is illustrated with two simple examples.
The representations of spl(2,1)—an example of representations of basic superalgebras
Authors: Marcu M.
Published: March 2003

The example of spl(2,1) illustrates some important properties of the representations of basic superalgebras. Typical representations are most similar to the representations of semisimple Lie algebras. We argue that nontypical representations are a degenerate type of representation. The structure of not fully reducible nontypical representations is discussed in detail. As opposed to semisimple Lie algebras, spl(2,1) also admits representations with nondiagonal Cartan subalgebra. There is a one-to-one correspondence between the representations of spl(2,1) and those of the generalized superalgebra osp(2,0,1,1). The osp(2,0,1,1) counterparts of nontypical irreducible representations of spl(2,1) are unfaithful.
Fock space representations of the Lie superalgebra A(0,n)
Authors: Palev Tchavdar D.
Published: March 2003

An infinite class of finite-dimensional irreducible representations and one particular infinite-dimensional representation of the special linear superalgebra of an arbitrary rank is constructed. For every representation an orthonormal basis in the corresponding representation space is found, and the matrix elements of the generators are calculated. The method we use is similar to the one applied in quantum theory to compute the Fock space representations of Bose and Fermi operators. For this purpose we first introduce a concept of creation and annihilation operators of a simple Lie superalgebra and give a definition of Fock-space representations.
The theory of spinors via involutions and its application to the representations of the Lorentz group
Authors: Kim Shoon K.
Published: March 2003

A simple theory of the spinor representations of the complex orthogonal group O(d,C) in the d-dimensional Euclidean space V(d) is presented via a basic lemma on involutional transformations and Cartan’s theorem on O(d,C). The arbitrary gauge factors of the representations are reduced to ± signs by introducing appropriate phase conventions. The concept of an axial involution is introduced. The plane rotations in V(d) are introduced and used to construct the representations of the proper orthogonal group O+(d,C). The Lorentz group is treated as a subgroup of O(4,C). The general expression for the basic 2×2 irreducible representations A(L0) of the proper orthochronous Lorentz group G(L0) is obtained by direct reduction of the 4×4 spinor representation S(L0) by means of the basic lemma on the involutional transformations. It is completely parameterized by the angle and the axis of the spacial rotation and by the velocity of the pure Lorentz transformation. The finite dimensional irreducible representations of the Lorentz group G(L) are discussed. The transformations of electro-magnetic field under G(L) are discussed in the most general form.
A global theory of supermanifolds
Authors: Rogers Alice
Published: March 2003

A mathematically rigorous definition of a global supermanifold is given. This forms an appropriate model for a global version of superspace, and a class of functions is defined which corresponds to superfields. This new construction is compared with several pre-existing definitions of supermanifold and graded manifold; it is shown to include all these definitions and to go beyond them, particularly in admitting the possibility of nontrivial topology in the anticommuting sector. Local differential geometry and potential applications to supergravity are considered.
Structure of the BBGKY hierarchy near phase transition
Authors: Ramanathan G. V. , Jedrzejek C.
Published: March 2003

A nonperturbative method, as opposed to diagrammatic expansions, is used to study critical phenomena in a fluid with a small hard core and a weak, long-ranged attractive potential. Using the natural small parameter related to the inverse of the range of the attractive potential, spatially uniformly valid asymptotic estimates are made for the magnitudes of all correlations (which are defined as the excess from the generalized superposition approximation) in a region near phase transition in arbitrary number of dimensions. It is shown that if the dimension of the space is larger than four, the correlation hierarchy truncates at the three-body level. The pair correlation satisfies a linear equation. The solution is precisely of Ornstein-Zernike form. For dimensions smaller than four, the hierarchy is still an infinite chain, but considerably simpler than the BBGKY hierarchy. In this case, at the critical point, the correlations are shown to satisfy a scaling law which is the same as that for S4 spin systems.
Large coupling expansions for eigenenergies and Regge trajectories of the general even-power potential with applications
Authors: Sharma L. K. , Choubey Jyoti , Müller-Kirsten H. J. W.
Published: March 2003

Large coupling expansions of eigenenergies, wave functions and Regge trajectories of the generalized even power potential V(r)=−g2𝒥∞j=0N2jr 2j are obtained. These general expansions are then used to obtain eigenenergy expansions and Regge trajectories for the anharmonic oscillator, Gauss, and similar potentials.
Multipoles and excitons in higher-order coherence
Authors: Weiszmann Andrei N.
Published: March 2003

Considering the coherence properties of an inelastically scattered quantum radiation by an ensemble of N identical three level atoms in a solid-like array, we addressed the question whether this process is actually mediated by a series of multipole interactions and/or a series of virtual intermediate-exciton transitions and how the type of the considered mediation is reflected in the higher order coherence properties of the scattered radiation. Using a projector technique the reduced field density operator, the atomic and field correlation functions were found, the higher order coherence properties were studied and the arrival-time and phase correlation conditions of the light quanta were obtained and analysed. The results suggest that not only the arrival-time and phase correlation conditions are mathematically independent from the type of the considered mediating process, but that the multipole interaction approximation of a resonant Raman scattering, in the above considered system, is physically equivalent with the virtual intermediate-exciton transitions too.
Jet bundles and path structures
Authors: Coleman R. A. , Korte H.
Published: March 2003

The analysis of path structures is formulated in terms of jet bundles with particular emphasis on the transformation laws and symmetry properties of geodesic path structures. The role played by geodesic path structures in the constructive axioms of Ehlers, Pirani, and Schild for GRT is discussed and it is shown that these axioms are decidable.
Feynman amplitude circuit breakers
Authors: Hampton Richard R.
Published: March 2003

It is shown that point splitting may be used to regularize arbitrary Feynman amplitudes and Taylor subtraction terms of the type found in the Zimmermann formulation (BPHZ) of renormalized perturbation theory. Point splitting refers to the Fourier transformation of the Feynman integrand with respect to the internal (loop) momenta. It is shown that this has the effect in configuration space of breaking open the circuits of the diagram. The regularization may be removed, interchangeably, before or after the propagator epsilon is taken to zero in the sense of distribution theory. This result is proven elsewhere but is mentioned here for its importance. The result may make the method useful in establishing the unitarity of BPHZ. Two appendices contain a review of circuit-based graph theory and a demonstration that an arbitrary circuit based graph may sometimes have no realization as a graph.
Shift-operator techniques for the classification of multipole-phonon states. I. Properties of shift operators in the R(5) group
Authors: Vanden Berghe G. , De Meyer H. E.
Published: March 2003

With a view to obtaining an orthogonal solution to the state labeling problem of the nuclear quadrupole-phonon states, independent operators which shift the eigenvalues of the R(3) Casimir operator L2 are constructed. Expressions which connect quadratic products of these shift operators with R(5) invariants are given.
Shift-operator techniques for the classification of multipole-phonon states. II. Eigenvalues of the quadrupole shift operator O0l
Authors: De Meyer H. E. , Vanden Berghe G.
Published: March 2003

By the aid of previously derived relations involving shift operators and their products, a set of rules is set up for the determination of eigenvalues of the scalar Hermitian shift operator O0l in the space of R(5) nuclear quadrupole-phonon states. The eigenvalues are listed for all seniority states with v<8.
Shift operator techniques for the classification of multipole-phonon states. III. The quadrupole-phonon eigenstates of O0l
Authors: De Meyer H. E. , Vanden Berghe G.
Published: March 2003

It is shown that the non-l-degenerated normalized eigenstates of O0l can be identified with previously constructed quadrupole-phonon states which have been labeled by the intermediate angular momentum values. The l-degenerated O0l eigenstates are proven to be identical to the corresponding eigenstates of the fifth label generating operator S. From the analysis of the relationship between O0l and S, a rigorous expression for S in terms of canonical operators is withdrawn, while it also becomes possible to obtain the S eigenvalues in exact irrational form.
On some solutions of the polaron model
Authors: Ringwood G. A. , Devreese J. T.
Published: March 2003

A paper by one of the present authors (JTD) and others obtained solutions of the classical polaron model in which the electron rotates about the stationary polaron center. The radius of gyration of the electron was, however, shown to diverge with the radius of the Brillouin zone. In this paper the analysis is carried out more extensively and concisely and it is found that there are solutions for which the radius of gyration remains finite.
Quadratic Hamiltonians in phase space and their eigenstates
Authors: Moshinsky M. , Winternitz P.
Published: March 2003

The solution of the Schrödinger equation for a Hamiltonian ℋ that is a general second order polynomial in the canonical coordinates qi and momenta pi is discussed. Examples of such Hamiltonians abound in various fields of physics and, e.g., the problem of small vibrations has been solved a long time ago. In the general case we first use linear canonical transformations [the group Sp (2n, R)] to reduce ℋ to a simplified representative for ℋR. Next we imbed each ℋR into a complete set of commuting second order integrals of motion, making use of a recently obtained classification of maximal Abelian subalgebras of the algebra sp (2n, R). This imbedding is then used to separate variables in the representative Schrödinger equations and to obtain complete sets of their eigenfunctions. Finally the solutions of the representative equations can be transformed back into those of the original one making use of representations of the canonical transformations. The program is implemented fully for n=1,2 and its structure is analyzed for arbitrary n. Special cases of interest are treated in detail, e.g., Hamiltonians for a system of n particles invariant under rotations, translations and permutations.
Exact eigenvalues, eigenfunctions, matrix elements and phase-shifts for a particle of angular momentum l in a particular screened potential
Authors: Myhrman Ulla
Published: March 2003

The radial Schrödinger equation is solved for an effective potential that can be considered as a generalization of a potential suggested by Hylleraas and Risberg and by Hulthén. For the scattering problem the phase-shifts are deduced. For the bound state problem the energy eigenvalues and the normalized eigenfunctions are derived. Closed-form expressions for certain matrix elements are calculated, and for the particular case of expectation values a recurrence formula is derived. In an appendix Levinson’s theorem is discussed for the particular potential under consideration.
Ermakov systems, nonlinear superposition, and solutions of nonlinear equations of motion
Authors: Reid James L. , Ray John R.
Published: March 2003

We report several important additions to our original discussion of Ermakov systems. First, we show how to derive the Ermakov system from more general equations of motion. Second, we show that there is a general nonlinear superposition law for Ermakov systems. Also, we give explicit examples of the nonlinear superposition law. Finally, we point out that any ordinary differential equation can be included in many Ermakov systems.
The anholonomic Cauchy problem in general relativity
Authors: Stachel John
Published: March 2003

The Lie derivative approach to the Cauchy problem in general relativity is applied to the evolution along an arbitrary timelike vector field for the case where the dynamical degrees of freedom are chosen as the (generally anholonomic) metric of the hypersurface elements orthogonal to the vector field. Generalizations of the shear, rotation, and acceleration are given for a nonunit timelike vector field, and applied to the three-plus-one breakup of the Riemann tensor into components parallel and orthogonal to the vector field, resulting in the anholonomic Gauss–Codazzi equations. A similar breakup of the Einstein field equations results in the form of the constraint and evolution equations for the anholonomic case. The results are applied to the case of a space–time with a timelike Killing vector field (stationary field) to demonstrate their utility. Other possible applications, such as in the numerical integration of the field equations, are mentioned. Definitions are given of three-index shear, rotation, and acceleration tensors, and their use in a two-plus-two decomposition of the Riemann tensor and field equations is indicated.
A solution of the Cauchy initial value problem in the nonsymmetric theory of gravitation
Authors: Moffat J. W.
Published: March 2003

The Cauchy initial value problem in a theory of gravity based on a nonsymmetric Hermitian metric is investigated. The field equations are solved in terms of a series expansion of the metric gμν. In the nth order the equations take the form of either second-order hyperbolic partial differential equations for gij (i, j=1,2,3) or a first-order equation for the vector gauge field Wi. Given initial data on a space-like surface S (x0=0), the integration forward in time of the equations can be performed, assuming that the field variables are reasonably smooth and analytic.
The condensed phase of the imperfect Bose gas
Authors: Fannes M. , Verbeure A.
Published: March 2003

Correlation inequalities are used to prove rigorously the following for the imperfect Bose gas: (i) existence of condensation in and only in the ground state, (ii) unicity and existence of the limiting Gibbs states.
A note on the symmetries of the 3j and 6j coefficients. II
Authors: Venkatesh K.
Published: March 2003

The partition of the symmetries of the 3j and 6j coefficients according to the possible expressions for the number of terms in their series representations are discussed. The canonical parametrization of the 3j and 6j coefficients introduced by Lockwood are also discussed.
Micu-type invariants of exceptional simple Lie algebras
Authors: Agrawala Vishnu K.
Published: March 2003

A formula simplifying the quartic trace in exceptional simple Lie algebras is derived and used for expressing higher order Micu-type invariants as polynomials in the second order Casimir invariant.
On the intrinsic geometry of certain nonlinear equations: The sine-Gordon equation
Authors: Chinea F. J.
Published: March 2003

The classical relation between two-dimensional spaces of constant curvature and certain nonlinear partial differential equations is formulated in group-theoretic terms by means of the underlying semisimple isometry group. Rather than working with the metric and curvature in the given constant curvature space, it is then possible to consider the equivalent system consisting of a pair of first-order partial differential equations in flat space for two so(2,1) vectors satisfying a pair of SO(2,1)-invariant algebraic constraints. Such equations determine a SL(2,R) principal bundle with flat connection. The construction is carried out in detail for the case of the sine-Gordon equation. The connection is shown to give rise to a spectral problem of the inverse scattering type by means of a gauge transformation. Bäcklund transformations are shown to be automorphisms of the connection characterized by a certain so(2,1) null vector. The generation of solutions of the nonlinear equation from known ones is seen to be determined by gauge transformations leaving invariant a fixed set of gauge conditions.
Bäcklund transformations in several variables
Authors: Payne Dean A.
Published: March 2003

A differential geometric method of constructing Bäcklund transformations for a second-order partial differential equation in n independent variables (n arbitrary) is developed and several examples worked out.
Elementary solutions of the linear transport equation for continuously varying spatial media
Authors: Pomraning G. C. , Larsen E. W.
Published: March 2003

We demonstrate that it is possible to construct elementary solutions (eigenfunctions) of the linear transport equation for certain types of continuously varying spatial media. In general, both discrete and continuum modes result, which appear to be complete on the half-range. A detailed analysis is given for an ’’exponential’’ medium, including numerical results and a half-range completeness proof. A ’’linear’’ medium is also considered. A general method is presented for constructing, jointly, the spatial variation of a medium and the corresponding functional forms of the eigenfunctions. Our results represent a partial generalization of the singular eigenfunction technique to media with continuous spatial variation.
Solution of the Chandrasekhar H-equation by Newton’s Method
Authors: Kelley C. T.
Published: March 2003

For any value of the parameter c, c≠1, for which a solution to the Chandrasekhar H-equation exists, Newton’s method may be used to compute the solution by iteration.
Generalized method of a resolvent operator expansion. III
Authors: Znojil M.
Published: March 2003

We show that our Born-like parametrized expansion of R(E)=(E−H)−1 is, after minor modifications, well defined even at the pole of R(E) and provides a new method of solving linear homogeneous equations. Three different possibilities of application are discussed here: (1) An analytic method of solving the differential eauations. It is based on the partitioning of generalized power series and illustrated by the new solution of the s-wave Schrödinger equation. (2) A consequent model space reduction of Schrödinger equation. In terms of the matrix moments of H, the effective interaction is defined as an operator continued fraction. (3) A new form of the perturbation theory which dispenses with the solution of the unperturbed problem.
Regularization of Hamiltonians and processes
Authors: Albeverio S. , Ho/egh-Krohn R. , Streit L.
Published: March 2003

For Markov processes and Hamiltonians given by energy forms, singular drift coefficients and potentials, more general than distributions, are allowed. We show that the forms can be regularized in such a way that the approximating processes and semigroups converge as the regularization is removed.
Absorption time by a random trap distribution
Authors: Rosenstock Herbert B.
Published: March 2003

Consider a particle performing a random walk to nearest neighbors on a simple cubic lattice in any number of dimensions D. The lattice contains a fraction q of randomly located ’’trapping’’ sites which absorb the walker when stepped on. We calculate the mean time to trapping. This involves the expected number V(t) of distinct sites that a walker would visit in t steps in absence of traps, a quantity known only asymptotically for large t; however, an exact calculation possible in one dimension suggests that the results are unexpectedly precise. The trapping time is proportional to q−1 for D=3 or greater, and more complicated for D=1,2.