bluuebunny/crossref_metadata_2025 huggingface dataset:
A description of virtual scattering states in the rigged Hilbert space formulation of quantum mechanics
Authors: Gadella M.
Published: March 2003

We define a pair of Gel’fand triplets Φi ⊆ℋi⊆Φ′i, i=1,2, where the linear spaces Φi are formed by suitably chosen entire analytic functions on the Riemann surface associated to the transformation ω=z2. Virtual scattering states are well defined as pairs of functionals, respectively, in Φ1 and Φ′2.
Inverse scattering by a local impurity in a periodic potential in one dimension
Authors: Newton Roger G.
Published: March 2003

Hill’s equation, modified by a potential that vanishes as x → ±∞, is considered. The direct scattering problem is studied; analytic and asymptotic properties of solutions of Hill’s equation as well as of solutions of the modified equation are established. A new version of Levinson’s theorem is proved. The inverse scattering problem is solved by means of a Marchenko-like equation.
A new class of integrable systems
Authors: Dorizzi B. , Grammaticos B. , Ramani A.
Published: March 2003

We present a family of dynamical systems associated with the motion of a particle in two space dimensions. These systems possess a second integral of motion quadratic in velocities (apart from the Hamiltonian) and are thus completely integrable. They were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation (PDE) for the potential. A most important point is that the same PDE was derived through considerations on the analytic structure of the singularities of the solutions (‘‘weak-Painlevé property’’).
A deformation of the general zero-curvature equations associated to simple Lie algebras
Authors: Kupershmidt Boris , Peterson Dale
Published: March 2003

We construct deformations and rational reductions for all the general zero-curvature equations associated to simple complex Lie algebras [known as AKNS equations for sl(2, C)].
Higher-order parabolic approximations to time-independent wave equations
Authors: Corones J. P. , Krueger R. J.
Published: March 2003

A sequence of numerically tractable higher-order parabolic approximations is derived for the reduced wave equation in an inhomogeneous medium. The derivation is motivated by a definition of waves propagating in a distinguished direction. For a homogeneous medium these definitions are exact and yield uncoupled, infinite-order parabolic equations which are equivalent to the wave equation. The difficulty of obtaining higher-order parabolic approximations for the elastic wave equation in an inhomogeneous medium is also discussed.
Simple waves in quasilinear hyperbolic systems. II. Riemann invariants for the problem of simple wave interactions
Authors: Grundland Alfred M. , Żelazny Roman
Published: March 2003

In this paper a generalization of the Riemann invariant method to the case of a nonhomogeneous system of equations has been formulated. We have discussed in detail the necessary and sufficient conditions for the existence of Riemann invariants. We perform the analysis using the apparatus of differential forms and Cartan theory of systems in involution. The problem has been reduced to examining Pfaff forms. We have considered the connections between the structure of the set of integral elements and the possibility of a construction of special classes of solutions depending on k arbitrary functions of one variable. These solutions can be interpreted physically as the interactions between k simple waves on a simple state. We have proven that, in the case of interaction of many simple waves described by Riemann invariants, a conservation law for the type and quantity of waves holds. It has been also shown that such a solution, resulting from the interaction of many simple waves propagating on the simple state, decay for a large time in an exact way into simple waves (of the same kind as those entering the interaction) on the state. The Cauchy problem for the nonlinear superposition of k-sample waves has been formulated. A couple of theorems useful for this problem have been given in the Sec. III. The functorial properties of the system of equations determining Riemann invariants have been described. The last part of the work contains an analysis of some examples of the solutions of nonhomogeneous magnetohydrodynamic equations from the point of view of the method described above.
Generalized stochastic processes and continual observations in quantum mechanics
Authors: Lupieri Giancarlo
Published: March 2003

We give here a mathematically rigorous form to an earlier work by Barchielli, Lanz, and Prosperi, in which it was found that a generalized stochastic process describes the results of continual observations of the position of a quantum particle. With the help of Albeverio and Ho/egh-Krohn’s theory of Feynman path integrals, we define the characteristic functional of this process and demonstrate that it possesses the necessary properties of normalization, continuity, and positive definiteness. An explicit calculation of the Feynman path integral which defines the functional allows an analysis of the process to be made.
A time-dependent Schrödinger equation
Authors: Kranold H. U.
Published: March 2003

We reduce the solution of the Schrödinger equation with the potential U(r, t)=α1(t) r2+α2(t) x+α3(t) y +α4(t) z+α5(t) to the solution of the Schrödinger equation for a free particle. The αi(t) are arbitrary functions of time. A generalization of this is also considered.
Lorentz covariance of an extended object in the tree approximation. II. Nonspherical object in 3+1 dimensions
Authors: Umezawa M.
Published: March 2003

This is the second in the series of the papers in which we investigate the Lorentz covariance of the extended object. In this paper we examine the covariance of the deformed object in 3+1 dimensions in the tree approximation. We construct the solution of the Euler equation, which is Lorentz covariant. In such a covariant solution, the variables associated with the rotational and the translational zero modes appear as classical quantum mechanical operators. Consequently the covariant solution has an intrinsic spin, in addition to the intrinsic quantum mechanical momenta. Then, at the end of this work we will show that such a covariant solution can be obtained also by quantizing a classical solution of the Euler equation, having extra variables signifying the center and the orientation of the deformed object. In the tree approximation, the energy–momentum and the relativistic angular momentum of the extended object ψ become pure classical quantum mechanical operators, having been integrated over the space. Then it is proven that such four-momenta and angular momentum operators form a classical quantum mechanics presented in a relativistic manner. The center of mass of the extended object, often called collective coordinate, is shown to be made of these four-momentum and angular momentum. This center of mass and the four-momentum operators form a quantum mechanics presented in the conventional form.
Unified geometrical approach to relativistic particle dynamics
Authors: Balachandran A. P. , Marmo G. , Mukunda N. , Nilsson J. S. , Simoni A. , Sudarshan E. C. G. , Zaccaria F.
Published: March 2003

Models for systems of relativistic particle dynamics are reviewed in terms of a geometrical setting for constraint dynamics. They are derived from the same grand abstract space by means of a common reduction procedure and are put in correspondence with invariant subgroups of the Poincaré group. A new model corresponding to the identity subgroup is also discussed.
A smooth transonic flow in the plane
Authors: Smith P. D.
Published: March 2003

The implicit function theorem is used to study a symmetric exterior problem for the gas dynamics equation—an equation of mixed type. The existence of families of smooth C1 solutions is demonstrated. These solutions are families of smooth transonic flows in the plane and are of applied interest. Some of these results have appeared in the literature with an incorrect derivation using the Hodograph mapping. This mapping is not invertible in the transonic case. The methods of this paper do not use the Hodograph mapping and extend to general (e.g., plasma) flows.
Decomposition of representations into basis representations for the classical groups
Authors: D’Hoker E.
Published: March 2003

We prove decomposition formulae for an arbitrary representation in terms of basis representations for the classical compact Lie groups. Using these decomposition formulae, simple rules are obtained for the product of two arbitrary representations and for the restriction of a representation to a classical subgroup.
Expansions over the ‘‘squared’’ solutions and difference evolution equations
Authors: Gerdjikov V. S. , Ivanov M. I. , Kulish P. P.
Published: March 2003

The completeness relation for the system of ‘‘squared’’ solutions of the discrete analog of the Zakharov–Shabat problem is derived. It allows one to rederive the known statements concerning the class of difference evolution equations related to this linear problem and to obtain additional results. These include: (i) the expansion of the potential and its variations over the system of ‘‘squared’’ solutions, the expansion coefficients being the scattering data and their variations, respectively; thus the interpretation of the inverse scattering transform (IST) as a generalized Fourier transform becomes obvious; (ii) compact expressions for the trace identities through the operator Λ, for which the ‘‘squared’’ solutions are eigenfunctions; (iii) brief exposition of the spectral theory of the operator Λ; (iv) direct calculation of the action-angle variables based on the symplectic form of the completeness relation; (v) the generating functional of the M operators in the Lax representation; (vi) the quantum version of the IST.
Hamiltonian operators with maximal eigenvalues
Authors: Harrell Evans M.
Published: March 2003

The potentials V(x) with a given Lp norm that maximize the lowest eigenvalue of −Δ+V are characterized.
Stochastic path-ordered exponentials
Authors: Potthoff Jürgen
Published: March 2003

We prove convergence of an approximation of the stochastic product integral for conditional Wiener paths to the solution of a certain stochastic integral equation. This is used to establish the Wiener–Itô representation for the kernel of the semigroup exp t ΔA, where ΔA =∑μ (∂μ1+Aμ)2 for functions Aμ with values in the space of anti-Hermitian matrices.
Path integrals in parametrized theories: Newtonian systems
Authors: Hartle James B. , Kuchař Karel V.
Published: March 2003

Constraints in dynamical systems typically arise either from gauge or from parametrization. We study Newtonian systems moving in curved configuration spaces and parametrize them by adjoining the absolute time and energy as conjugate canonical variables to the dynamical variables of the system. The extended canonical data are restricted by the Hamiltonian constraint. The action integral of the parametrized system is given in various extended spaces: Extended configuration space or phase space and with or without the lapse multiplier. The theory is written in a geometric form which is manifestly covariant under point transformations and reparametrizations. The quantum propagator of the system is represented by path integrals over different extended spaces. All path integrals are defined by a manifestly covariant skeletonization procedure. It is emphasized that path integrals for parametrized systems characteristically differ from those for gauge theories. Implications for the general theory of relativity are discussed.
Quantum measuring processes of continuous observables
Authors: Ozawa Masanao
Published: March 2003

The purpose of this paper is to provide a basis of theory of measurements of continuous observables. We generalize von Neumann’s description of measuring processes of discrete quantum observables in terms of interaction between the measured system and the apparatus to continuous observables, and show how every such measuring process determines the state change caused by the measurement. We establish a one-to-one correspondence between completely positive instruments in the sense of Davies and Lewis and the state changes determined by the measuring processes. We also prove that there are no weakly repeatable completely positive instruments of nondiscrete observables in the standard formulation of quantum mechanics, so that there are no measuring processes of nondiscrete observables whose state changes satisfy the repeatability hypothesis. A proof of the Wigner–Araki–Yanase theorem on the nonexistence of repeatable measurements of observables not commuting conserved quantities is given in our framework. We also discuss the implication of these results for the recent results due to Srinivas and due to Mercer on measurements of continuous observables.
The Darboux transformation and solvable double-well potential models for Schrödinger equations
Authors: Zheng W. M.
Published: March 2003

The Darboux transformation, a method used to transform a Schrödinger-type equation to a Schrödinger equation with a new potential, is discussed. An exactly solvable double-well potential model for the one-dimensional Schrödinger equation is obtained.
Eigenvalues and eigenfunctions associated with the Gel’fand–Levitan equation
Authors: Moses Harry E. , Prosser Reese T.
Published: March 2003

It is shown here that the solutions of the Gel’fand–Levitan equation for inverse potential scattering on the line may be expressed in terms of the eigenvalues and eigenfunctions of certain associated operators of trace class. The details are sketched for the case of rational reflection coefficients, and carried out for the simplest class of examples.
The causal automorphism of de Sitter and Einstein cylinder spacetimes
Authors: Lester J. A.
Published: March 2003

A well-known result, due originally to Alexandrov in 1953 and subsequently rediscovered by Zeeman in 1964, states that transformations of Minkowski spacetime which preserve causality are essentially orthochronous Lorentz transformations. In this article, we first exhibit a proof of this result by using a lemma of Zeeman to reduce the proof to another well-known theorem of Alexandrov involving transformations preserving light speed. Then, by generalizing Zeeman’s lemma and using recent extensions of Alexandrov’s light-speed theorem, we determine the causal automorphisms of de Sitter and Einstein cylinder spacetimes.
Spherically symmetric solution in the nonsymmetric Kaluza–Klein theory
Authors: Kalinowski M. W. , Kunstatter G.
Published: March 2003

In this paper we find an exact, static, spherically symmetric solution for the nonsymmetric Kaluza–Klein theory. This solution has the remarkable property of describing ‘‘mass without mass’’ and ‘‘charge without charge.’’ We examine its properties and a physical interpretation.
On classes of integrable systems and the Painlevé property
Authors: Weiss John
Published: March 2003

The Caudrey–Dodd–Gibbon equation is found to possess the Painlevé property. Investigation of the Bäcklund transformations for this equation obtains the Kuperschmidt equation. A certain transformation between the Kuperschmidt and Caudrey–Dodd–Gibbon equation is obtained. This transformation is employed to define a class of p.d.e.’s that identically possesses the Painlevé property. For equations within this class Bäcklund transformations and rational solutions are investigated. In particular, the sequences of higher order KdV, Caudrey–Dobb–Gibbon, and Kuperschmidt equations are shown to possess the Painlevé property.
Properties of the Schwinger model
Authors: Capri Anton Z. , Ferrari Ruggero
Published: March 2003

We present all the Wightman functions for an explicit operator solution of the Schwinger model. To understand these better, we study the algebra of fields of this model, representations of this algebra as well as the Hamiltonian. The latter turns out to elucidate the ‘‘confinement’’ of the fermion field. In addition we comment on the renormalization of the theory as well as on the analyticity of the amplitudes in terms of the coupling constant.
A novel mass-eigenvalue problem for spinors in deSitter space
Authors: Kerner Edward H.
Published: March 2003

It is shown that an unambiguous quantum theory of spinors in positively curved deSitter space, based on distinguished coordinates in a Hamiltonian framework, leads to a set of spinors corresponding to unsharp energy but sharp mass defined in a family of novel eigenvalue problems. An example is given in which partly real and partly complex discrete mass spectra come forth.
The connection between time- and frequency-domain three-dimensional inverse scattering methods
Authors: Rose James H. , Cheney Margaret , DeFacio Brian
Published: March 2003

We relate three-dimensional scattering theory for the time-independent Schrödinger equation without spherical symmetry to scattering theory for the plasma-wave equation (PWE). We review a recent inverse scattering method for the PWE and find the corresponding method for the Schrödinger equation. We then review Newton’s three-dimensional Marchenko method for the Schrödinger equation and transform it to the corresponding PWE method. The resulting time-domain hyperbolic method clarifies the role of causality in Newton’s important recent work.
Aspects of high-dimensional theories in embedding spaces
Authors: Maia M. D. , Mecklenburg W.
Published: March 2003

In this paper we concern ourselves with the question of whether physical meaning may be attributed to the extra dimensions provided by embedding procedures applied to physical space-times. We also comment on similarities and differences to conventional Kaluza–Klein pictures.
Higgs fields from symmetric connections—The bundle picture
Authors: Hudson L. B. , Kantowski R.
Published: March 2003

The derivation (Forgács and Manton) of a classical gauge plus Higgs field theory from a purely symmetric gauge theory by dimensional reduction is translated into the pictorial language of symmetric connections on principal bundles. The bundle description provides a better understanding of symmetric gauge theories; its global nature excludes some gauge group-symmetry group combinations, and shows that the reduced gauge group is larger than the centralizer CH as previously thought. In general it is a certain factor group, NK/K, involving both the gauge and symmetry groups, and contains CH as a normal subgroup. The commonly used F2 Lagrangian of symmetric gauge theories will not prescribe the dynamics of fields associated with this larger group, unless these fields are suitably constrained. It is shown that a constrained Kaluza–Klein metric is required to construct the F2 Lagrangian, thus pointing the way to include dynamics for the NK/K group.
An asymptotic behavior of the S-matrix and the inverse scattering problem
Authors: Saitō Yoshimi
Published: March 2003

An asymptotic formula of the S-matrix as k→∞ will be shown for the general short-range scattering in RN(N≥2). The main tool for the proof is the spectral decomposition theorem for the Schrödinger operator. By the use of the asymptotic formula, the uniqueness of the inverse scattering problem will be shown and a reconstruction formula for the potential will be presented for the general short-range scattering.
A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. II
Authors: Ralston James
Published: March 2003

This article continues the analysis of the preceding article, showing that the Lyapunov functional introduced by Newman can be used to prove the stability of the Barenblatt–Pattle solutions of the porous medium equation.
Enveloping algebra annihilators and projection techniques for finite-dimensional cyclic modules of a semisimple Lie algebra
Authors: Gould M. D. , Edwards S. A.
Published: March 2003

Some results on the structure of finite-dimensional cyclic modules for a semisimple Lie algebra are presented. Cyclic modules arise naturally in constructing symmetry adapted states of a system using projection. Projecting out states with definite symmetry from an arbitrary state ψ is related to the properties of the cyclic module generated by ψ. An important example of a cyclic module is the tensor product of two irreducible modules V(λ)⊗V(μ) which is cyclically generated by the vector vλ−⊗vμ+, where vλ−(resp., vμ+) is the minimal (resp., maximal) weight vector of V(λ) [resp., V(μ)]. For this particular case we determine the explicit form of the annihilator, in the universal enveloping algebra, of the cyclic vector vλ−⊗vμ+. It is hoped that this result may add new insight into the Clebsch–Gordan multiplicity problem. As an application of this result projection operators are constructed which project, from an arbitrary vector of weight λ, a maximal weight vector of weight λ.
Differential equations for the cuspoid canonical integrals
Authors: Connor J. N. L. , Curtis P. R.
Published: March 2003

Differential equations satisfied by the cuspoid canonical integrals In(a) are obtained for arbitrary values of n≥2, where n−1 is the codimension of the singularity and a=(a1,a2,...,an−1). A set of linear coupled ordinary differential equations is derived for each step in the sequence In(0,0,...,0,0) →In(0,0,...,0,an−1) →In(0,0,...,an−2,an−1) →...→In(0,a2,...,an−2,an−1) →In(a1,a2,...,an−2,an−1). The initial conditions for a given step are obtained from the solutions of the previous step. As examples of the formalism, the differential equations for n=2 (fold), n=3 (cusp), n=4 (swallowtail), and n=5 (butterfly) are given explicitly. In addition, iterative and algebraic methods are described for determining the parameters a that are required in the uniform asymptotic cuspoid approximation for oscillating integrals with many coalescing saddle points. The results in this paper unify and generalize previous researches on the properties of the cuspoid canonical integrals and their partial derivatives.
The Haar integral for Lie supergroups
Authors: Williams D. , Cornwell J. F.
Published: March 2003

Each supermanifold can be considered as equivalent to a certain family of real manifolds. The Haar integral of a Lie supergroup is defined using this equivalence. A useful general formula is derived together with explicit construction methods of the simple Lie supergroups of types SPL(n‖m;EL), OSP(n‖2r;EL), and B(n;EL), and the extended Poincaré group.
A novel class of Bessel function integrals
Authors: Glasser M. L.
Published: March 2003

A number of definite integrals over the unit interval involving Bessel functions with argument [α2x−1+β2(1−x)−1]1/2 are evaluated in closed form.
The shift operator technique for SO(7) in an [SU(2)]3 basis. I. Theory
Authors: Van der Jeugt J. , De Wilde P.
Published: March 2003

Shift operators for the chain of Lie groups SO(7)⊇[SU(2)]3 are constructed, and several properties are examined. Quite a number of relations connecting quadratic products of shift operators are built up. Finally, the connection between matrix elements of the shift operators and reduced matrix elements of the tensor operator, which forms part of the SO(7) generator basis, is explained.
Formal relations between classical superalgebras and fermion–boson creation and annihilation operators
Authors: Tang Dun-Sang
Published: March 2003

Classical Lie superalgebras A(m−1, n−1), B(m, n), C(n), and D(m, n) are realized by sets of fermionic and bosonic creation and annihilation operators. Some aspects of the physical representations are discussed.
A note on general involutional transformations
Authors: Madivanane S.
Published: March 2003

For two matrices A and B of order n×n which satisfy xn−1=0, it is shown that the transformation matrix TAB which connects A and B via ATAB=TABB becomes general involutional when AB=ωBA, ω being a primitive nth root of unity.
Simultaneity and reality of U(n) and SU(n) 3j
m and 6j symbols
Authors: Bickerstaff R. P.
Published: March 2003

A simple proof is given of the simultaneous existence of a real set of 3jm and 6j symbols for the unitary groups U(n) and SU(n); including the case of mixed tensor representations. We observe that simultaneity is incompatible with the conventional permutation symmetries for U(n) with n>3. The relevance of these results to the Schur–Weyl duality of U(n) with the symmetric groups Sl is discussed.
General charge conjugation operators in simple Lie groups
Authors: Moody R. V. , Patera J.
Published: March 2003

A description of particular elements (‘‘charge conjugation operators’’) found in any compact simple Lie group K is presented. Such elements Ri transform a physical state (weight vector of a basis of a representation space) into others with opposite ‘‘charge (ith component of the weight), sometime changing also the sign of the state. It is demonstrated that exploitation of these elements and the finite subgroup N of K generated by them offer new powerful methods for computing with representations of the Lie group. Their application to construction of bases in representation spaces is considered in detail. It represents a completely new direction to the problem.
Clebsch–Gordan coefficients for E6 and SO(10) unification models
Authors: Koh In-Guy , Patera J. , Rousseau C.
Published: March 2003

We illustrate here a new method for computing Clebsch–Gordan coefficients (CGC) for E6 by computing CGC for the product 27⊗27 of the irreducible representation (100000) of E6 with itself. These CGC are calculated thrice: once in a weight vector basis independent of any semisimple subgroup, then in a basis which refers to SO(10)⊆E6, and finally in a basis referring to SU(5)⊆SO(10)⊆E6.
On the Fourier series representations of path integrals
Authors: Royer Antoine
Published: March 2003

The direct transformation from the polygonal to the Fourier series representation of Feynman path integrals, via a change of integration variables, is effected explicitly for cases where the Lagrangian is of the form L(x,ẋ,t)= 1/2 ẋ2−V(x,t). This transformation involves a ‘‘functional Jacobian’’ stemming solely from the velocity term in the Lagrangian; this is because N-segment polygonal paths of the type contributing significantly to the integral, and their N-term Fourier series approximants, coalesce together as N→∞, but not their derivatives. We also consider integrals over paths with fixed means T−1∫T0 dt x(t) =const. The usefulness of the Fourier representation is illustrated with the harmonic oscillator case V(x)= 1/2 ω2x2, in both the free and fixed means situations; in particular, the Fourier evaluation of the path integrals trivially determines the large time phases (Maslov indices), and the ranges of ω2 values for which the integrals are finite or infinite in the imaginary time cases.
Casimir invariants for the eight-dimensional subgroups of the Poincaré group P (1,4)
Authors: Leveille M.
Published: March 2003

The generalized Casimir operators of all eight-dimensional subgroups of the Poincaré group P(1,4) are found. These operators include polynomials in the enveloping algebra of the considered eight-dimensional Lie algebra, quotients of such polynomials, and also more complicated functions of the infinitesimal operators.
Finite- and infinite-dimensional representations of the orthosymplectic superalgebra OSP(3,2)
Authors: Van der Jeugt J.
Published: March 2003

The shift operator technique is used to give a complete analysis of all finite- and infinite-dimensional irreducible representations of the orthosymplectic superalgebra osp(3,2). For all cases, the star or grade star conditions for the algebra are investigated. Only two finite-dimensional representations are grade star representations, if the representation space is required to be a graded Hilbert space. When the even part is so(3)⊕sp(2,R)≊su(2)⊕su(1,1), an infinite class of infinite-dimensional star representations is found. One of them can be realized in terms of two-valued functions of a complex variable. This representation reduces to the sum of two metaplectic representations of sp(2). We show that it is precisely this ‘‘metaplectic representation for osp(3,2)’’ which gives the spin-energy eigenstates for the one-dimensional harmonic oscillator with spin 1/2 states.
Factorization of the 2×2 matrix recursion operator of the coupled KdV equation
Authors: Aiyer Raju N.
Published: March 2003

The recursion operator for the infinitesimal transformations about solutions of the coupled KdV equation is a 2×2 matrix whose elements are of the fourth order. This formidable looking operator is written as the product of four 2×2 matrix operators whose elements are of the first order. Auxiliary functions introduced to factorize the recursion operator lead to the scattering equation for the equation. The factorization of the recursion operator for the sine–Gordon equation is also presented.
Inverse scattering for geophysical problems. II. Inversion of acoustical data
Authors: Ramm A. G. , Weglein Arthur
Published: March 2003

An algorithm is given for finding the density and bulk modulus (refraction coefficient) of an inhomogeneity from the knowledge of the scattered field on the surface of the earth for all positions of the source and receiver on this surface and for two arbitrary fixed frequencies in the Born approximation. An alternative inversion method using the low-frequency data is also given.
Proof of Regge analyticity for power law potentials
Authors: Datta D. P. , Mukherjee S.
Published: March 2003

The l-plane analyticity of Schrödinger energy levels En(l) for power law potentials V(r)=rα, for α>2 has been proved by using the Kato–Rellich perturbation theory for linear operators.
Functions analytic on the half-plane as quantum mechanical states
Authors: Paul Thierry
Published: March 2003

A transform between the state space of one-dimensional quantum mechanical systems and a direct sum of two spaces of square integrable functions analytic on the open upper half-plane is constructed. It gives a representation of usual quantum mechanics on which the free evolution is trivial and the representation of canonical transformation very simple. Generalizations to higher dimensions are also discussed.
Quantum systems with time-dependent harmonic part and the Morse index
Authors: Rezende Jorge
Published: March 2003

A simple relation between two quantum systems with a time-dependent, respectively, time-independent, harmonic part is established. Using this we give a computation, valid for all times, of the Green’s functions of the time-dependent harmonic oscillator with and without a perturbation of the type g/x2. The asymptotic expansion of the wave function in powers of Planck’s constant is discussed using a new representation of the Morse index.
Kantowski–Sachs cosmological models approaching isotropy
Authors: Weber Edmond
Published: March 2003

In this article we analyze a particular class of anisotropic cosmological models, the Kantowski–Sachs models, in the presence of a nonzero cosmological constant Λ. We study them qualitatively by means of autonomous systems with two and three dimensions. The plane autonomous system gives a new class of empty Kantowski–Sachs cosmologies, with Λ>0 and Λ<0. We find two new types of singularity points. The autonomous system with three dimensions yields a set of solutions of nonzero measure becoming isotropic in an infinite cosmological time.
Statistical mechanics of the gravitational field in a conformally static setting
Authors: Kandrup Henry E.
Published: March 2003

This paper formulates a statistical description of a collection of N identical classical particles that interact relativistically via linear fields in a fixed background space-time that admits a conformal timelike Killing field. Attention focuses upon the special cases of a simple scalar interaction and a linearized gravitation interaction which should suffice to model many systems of astrophysical interest. The fundamental object of the theory is a complicated distribution function that depends upon appropriate variables for both the particles and the fields. By assuming that, in a first approximation, this distribution factorizes into an infinite product of reduced distribution functions, one recovers the type of mean-field theory developed by such authors as Ipser and Thorne. Alternatively, one may derive various exact and approximate relations which contain information about the interparticle correlations.
Gribov copies and the Faddeev–Popov formula in lattice gauge theories
Authors: Sharpe Bruce
Published: March 2003

We present a rigorous, nonperturbative derivation of a lattice version of the Faddeev–Popov integral. This derivation shows that Gribov copies can occur in the lattice theory for certain gauges, but these copies do not affect normalized functional integrals in the lattice theory. Furthermore, taking the formal limit as the lattice spacing tends to zero leads to the usual continuum Faddeev–Popov integral.
Reduction formulas for higher-order indices and anomaly
Authors: Fujimoto Yasushi , Grigjanis Robert
Published: March 2003

Reduction formulas for {1k} and {k} of SU(N) are derived which are useful for obtaining higher-order indices. We also touch on the anomaly in higher-dimensional theories.
Space groups and their isotropy subgroups
Authors: Jarić Marko V. , Senechal Marjorie
Published: March 2003

The theory of space groups and their subgroups is related to that of their irreducible representations. Several important theorems concerning the determination of isotropy subgroups are proved. In particular, we show that the translation subgroups of isotropy subgroups are in one-to-one correspondence with certain subsets, called ‘‘substars,’’ of the ‘‘star’’ characterizing the irreducible representation of the space group.
Classification of systems of nonlinear ordinary differential equations with superposition principles
Authors: Shnider S. , Winternitz P.
Published: March 2003

A system of n first-order nonlinear ordinary differential equations ẋ(t)=f(x,t) is said to admit a superposition principle if its general solution can be written as a function of a finite number m of particular solutions and n constants. Such a system can be associated with the nonlinear action of a Lie group G on a space M. We show that ‘‘indecomposable’’ systems of ODE’s with supersposition principles are obtained if and only if the Lie algebras L0⊆L, corresponding to the isotropy group H of a point and the group G, respectively, define a transitive primitive filtered Lie algebra (L,L0). Using known results from the theory of transitive primitive Lie algebras we deduce that L0 must be a maximal subalgebra of L and that G must be an affine group, a simple Lie group, or the direct product of two identical simple Lie groups. Affine groups lead to linear equations, the other types to nonlinear equations with polynomial or rational nonlinearities. Equations corresponding to the classical complex Lie algebras are worked out in detail.
Necessary conditions for a unique solution to two-dimensional phase recovery
Authors: Barakat Richard , Newsam Garry
Published: March 2003

In this paper we show that although in one dimension multiplicity of solutions to the phase reconstruction problem presents a serious problem, in two or more dimensions multiplicity is pathologically rare. We derive from a given solution pair (g,G) necessary conditions for the existence of alternative solution pairs (h,H), and a characterization of their form. The mathematical tools employed are from the theory of functions of two complex variables.
Commutants of a family of operators on a partial inner product space
Authors: Shabani J.
Published: March 2003

We consider all different possible definitions of commutants and bicommutants for an x-invariant family of operators on a partial inner product space. We investigate their behavior with respect to the weak topology and we describe the situation when all commutants (resp. all bicommutants) coincide.
Holonomy groups, sesquidual torsion fields, and SU(8) in d=11 supergravity
Authors: Castellani L. , D’Auria R. , Fré P. , van Nieuwenhuizen P.
Published: March 2003

The torsion and its curl form an anti-self-dual SO(8) tensor field FABCD( y). The Maxwell equations are solved if this tensor is covariantly constant, while the Einstein equations are solved if it satisfies an algebraic relation at the origin. Such FABCD are found as the kernel of the holonomy group in the 35 representation of SO(8) and they extend the SO(8) of the round S7 to SU(8).
Triangular Ising model: Complete expressions for even spin correlations in an exact evaluation of P(h,T)
Authors: Choy T. C.
Published: March 2003

Complete expressions of all even correlations for the triangular Ising model in a recent exact evaluation of P(h,T), the local magnetic field distribution, are obtained from the Pfaffian representation derived by Stephenson. The results for the ferromagnetic J<0 and antiferromagnetic J>0 cases contain a number of interesting features previously not pointed out in the Ising literature. As a bonus, some new inequalities relating odd and even correlations are also obtained.
Generalized logarithmic Borel summability
Authors: Grecchi V. , Maioli M.
Published: March 2003

The recently introduced logarithmic Borel summation method is able to sum strongly divergent series of a particular type. A satisfactory extension to the applicability of this method, obtained by using the classical Borel–Le Roy transform, is presented. As examples we consider a class of nonpolynomial anharmonic oscillator models in the ’t Hooft simplified form.
Two theorems on star diagrams
Authors: Houard J. C. , Irac-Astaud M.
Published: March 2003

The notion of star diagram, previously introduced for the study of Green functions of nonlinear differential operators is formulated in an algebraic frame. Two theorems are presented which make the structure of these functions explicit.
Formulas for the eigenvalues of the Laplacian on tensor harmonics on symmetric coset spaces
Authors: Pilch K. , Schellekens A. N.
Published: March 2003

On a symmetric coset space G/H the eigenvalues of the Laplacian and the Lichnerowicz operator acting on arbitrary tensor harmonics are given in terms of the eigenvalues of the quadratic Casimir operators of G and H. Explicit examples for Sn, CPn, and real (complex) Grassmann manifolds are analyzed.
Lie groups and Lie algebras with generalized supersymmetric parameters
Authors: Kobayashi Yuji , Nagamachi Shigeaki
Published: March 2003

Matrices with σ-symmetric parameters (the most general extension of supersymmetric parameters) are investigated. The superdeterminants of such matrices are defined. Lie groups consisting of these matrices and their Lie algebras are studied.
Determination of point group harmonics for arbitrary j by a projection method. III. Cubic group, quantization along a ternary axis
Authors: Raynal Jacques , Conte Robert
Published: March 2003

The method described in a first paper to obtain cubic harmonics quantized on an axis of order 4 is applied to the case of a ternary quantization axis. Projectors on irreducible representations are expressed with rotation matrices R(0, φ, π) and R(π, π-φ, 0), φ=arccos(1/3), acting on subspaces of SU(2) invariant under D3. Relations between the descriptions on the two axes of quantization are derived.
A nilpotent prolongation of the Robinson–Trautman equation
Authors: Glass E. N. , Robinson D. C.
Published: March 2003

A prolongation is constructed, in the sense of Wahlquist and Estabrook, for the nonlinear evolution equation determining Robinson–Trautman space-times. The Lie algebra so obtained is found to be (naturally) seven-dimensional and nilpotent. Representations of the algebra are considered. The simple relationship of such a prolongation to the conservation laws associated with the Robinson–Trautman equation is discussed.
Factorization method and new potentials with the oscillator spectrum
Authors: Mielnik Bogdan
Published: March 2003

A one-parameter family of potentials in one dimension is constructed with the energy spectrum coinciding with that of the harmonic oscillator. This is a new derivation of a class of potentials previously obtained by Abraham and Moses with the help of the Gelfand–Levitan formalism.
The double cnoidal wave of the Korteweg–de Vries equation: An overview
Authors: Boyd John P.
Published: March 2003

Earlier work of the author on the spatially periodic solutions of the Korteweg–de Vries equation is here extended via an in-depth treatment of a special case. The double cnoidal wave is the simplest generalization of the ordinary cnoidal wave discovered by Korteweg and de Vries in 1895. In the limit of small amplitude, the double cnoidal wave is the sum of two noninteracting linear sine waves. In the oppositie limit of large amplitude, it is the sum of solitary waves of two different heights repeated periodically over all space. Although special, the double cnoidal wave is important because it is but the particular case N=2 of a broad family of solutions known variously as ‘‘N-polycnoidal waves,’’ ‘‘finite gap,’’ ‘‘finite zone’’ solutions, ‘‘waves on a circle,’’ or ‘‘N-phase wave trains.’’ It has been shown by others that the set of N-polycnoidal waves gives the general initial value solution to the Korteweg–de Vries equation. This present work is the core of a three-part treatment of the double cnoidal wave. This part, the overview, presents graphic examples in all the important parameter regimes, explains how collision phase shifts alter the average speed of the two wave phases from the ‘‘free’’ velocities of the two solitary waves, describes the different branches or modes of the double cnoidal wave (it is possible to have many solitary waves on each spatial period provided they are of only two distinct sizes), and contrasts the results of this work with the very limited numerical calculations of previous authors. The second part describes how the problem of numerically calculating the double cnoidal wave can be reduced down to solving four algebraic equations by perturbation theory. The third part explains how the so-called ‘‘modular transformation’’ of the Riemann theta functions is important in interpreting N-polycnoidal waves.
Perturbation series for the double cnoidal wave of the Korteweg–de Vries equation
Authors: Boyd John P.
Published: March 2003

By means of the theorems proved earlier by the author, the problem of the double cnoidal wave of the Korteweg–de Vries equation is reduced to four algebraic equations in four unknowns. Two of the unknowns are the nonlinear phase speeds c1 and c2. Another is a physically irrelevant integration constant. The fourth unknown is the off-diagonal element of the symmetric, 2×2 theta matrix, which in turn gives the explicit coefficients of the Riemann theta function. The double cnoidal wave u(x,t) is then obtained by taking the second x-derivative of the logarithm of the theta function. Two separate forms of these four nonlinear ‘‘residual’’ equations are given. One is obtained from the Fourier series of the theta function and is useful for small wave amplitude. The other is based on the Gaussian series of the theta function and is highly efficient in the large amplitude regime where the double cnoidal wave is the sum of two solitary waves. Both sets of residual equations can be solved via perturbation theory and results are given to fourth order in the Fourier case and second order in the Gaussian case. The Gaussian-based perturbation series has the remarkable property that it converges more and more rapidly as the wave amplitude increases; the zeroth-order solution is the familiar double solitary wave. Numerical comparisons show that the two complementary perturbation series give accurate results in all the important regions of parameter space. (The ‘‘unimportant’’ regions are those in which the double cnoidal wave is an ordinary cnoidal wave subject to a very weak perturbation.) This is turn implies that even for moderate wave amplitude where the nonlinear interactions are not weak, and yet the solitary wave peaks are not well separated, at least to the eye, it is still qualitatively legitimate to describe the double cnoidal wave as either the sum of two sine waves or of two solitary waves of different heights.
The special modular transformation for polycnoidal waves of the Korteweg–de Vries equation
Authors: Boyd John P.
Published: March 2003

The modular transformation of the Riemann theta function is used to show that the implicit dispersion relation for the N-polycnoidal waves of the Korteweg–de Vries equation has a countable infinity of branches for N≥2. Although the transformation also implies that each branch or mode can be written in a countable infinity of ways, it is also shown that there is a unique ‘‘physical’’ representation for each mode such that the parameters of the theta function can be interpreted as wavenumbers and amplitudes in the limit of either very small or very large amplitude. Unfortunately, the small amplitude ‘‘physical’’ representation is different (by a modular transformation) from the large amplitude ‘‘physical’’ representation for a given mode, but this difference explains an apparent paradox as described in the text. The general modular transformation expresses the theta function in terms of complex wavenumbers, phase speeds, and coordinates that have no physical relevance to the Korteweg–de Vries equation, but it is shown that for N≥2, there is a subgroup, here dubbed the ‘‘special modular transformation,’’ which gives a real result. This subgroup is explicitly constructed for general N and presented as a table for N=2.
A rule for the total number of topologically distinct Feynman diagrams
Authors: Battaglia Franco , George Thomas F.
Published: March 2003

A rule for the total number of topologically distinct Feynman diagrams is presented for the ground state of a system of many identical particles interacting via a two-body potential.
Summation of strongly divergent perturbation series
Authors: Arteca Gustavo A. , Fernández Francisco M. , Castro Eduardo A.
Published: March 2003

A new method for summing strongly divergent perturbation series is presented. It is based on the change of the power series into a convergent sequence by means of an order-dependent mapping obtained from a simple scaling relation. The perturbation expansions for a one-dimensional integral and for the ground states of the anharmonic oscillator and of the linear confining potential model are accurately summed in the most unfavorable strong-coupling limit.
On charges of massless particles
Authors: L/opuszański Jan T.
Published: March 2003

We are concerned with the theorem of Weinberg and Witten stating that a massless particle of helicity ‖h‖> 1/2 cannot be a carrier of a charge of an internal symmetry induced by a Lorentz covariant current and that for a massless field theory of ‖h‖>1 a Lorentz covariant energy momentum tensor cannot be constructed. We complete the proof of the theorem, as that given by Weinberg and Witten it is inconclusive. We suggest how to evade the difficulties which arise as a consequence of this theorem.
On the local GL(4, R) gauge symmetry of hyperbolic complex metrics
Authors: Zhong Zai-Zhe
Published: March 2003

There exists a construction of hyperbolic complex Lorentz frames for any signature η. We prove GL(4, R) to be locally isomorphic to U(η, H) by Lie algebra theory. Therefore, the local GL(4, R) gauge symmetry of hyperbolic complex metric g in the nonsymmetric gravitational theory, in fact, is a symmetry of rotations of the hyperbolic complex Lorentz frames.
A spin- 1/2 particle formalism in curved space–time
Authors: Castagnino Mario A. , Chimento Luis , Harari Diego D. , Núñez Carmen A.
Published: March 2003

A formalism is presented to solve the Dirac equation in curved space–time and the spinorial solution is obtained. A general method is introduced to compute the particle creation. The kernel S1(x,x′) leading to the particle model is built out of considerations on the Minkowskian limit and high energy behavior.