bluuebunny/crossref_metadata_2025 huggingface dataset:
On irreducible endospectral graphs
Authors: Knop Jan V. , Müller Wolfgang R. , Szymanski Klaus , Trinajstić Nenad , Kleiner Alexander F. , Randić Milan
Published: July 2002

The paper considers endospectral trees, a special class of graphs associated with the production of numerous isospectral graphs. Endospectral graphs have been considered in the literature sporadically (the name was suggested very recently [M. Randić, SIAM J. Algebraic Discrete Meth. 6, 145 (1985)]). They are characterized by the presence of a pair of special vertices that, if replaced by any fragment, produce an isospectral pair of graphs. Recently Jiang [Y. Jiang, Sci. Sin. 27, 236 (1984)] and Randić and Kleiner (M. Randić and A. F. Kleiner, ‘‘On the construction of endospectral trees,’’ submitted to Ann. NY Acad. Sci.) considered alternative constructive approaches to endospectral trees and listed numerous such graphs. The listing of all such trees having n=16 or fewer vertices has been undertaken here. It has been found that relatively few endospectral trees have novel structural features and cannot be reduced to some already known endospectral tree. These few have been named ‘‘irreducible endospectral trees.’’ They are responsible for the occurrence of a large number of isospectral trees, leading to, when one considers trees of increasing size, the situation that led Schwenk [A. J. Schwenk, in New Directions in the Theory of Graphs, edited by F. Harary (Academic, New York, 1973), pp. 275–307] to conclude that ‘‘almost all trees are isospectral.’’
A note on topology of supermanifolds
Authors: Matsumoto Shuichi , Kakazu Kiyotaka
Published: July 2002

A topological structure on the space of supernumbers is introduced, with which this space becomes a Fréchet space. The definition of supermanifold and ‘‘superprojective space’’ are given. The superprojective space is one of paracompact Hausdorff supermanifolds.
Collectivity and geometry. V. Spectra and shapes in the two-dimensional case
Authors: Chacón E. , Hess P. O. , Moshinsky M.
Published: July 2002

In the present paper a program for a collective symplectic model in a two-dimensional space based on the sp(4,R) Lie algebra is developed in full. The problem is only of conceptual interest but, as it admits a complete analytic discussion, it provides insights into more realistic collective nuclear models such as the interacting boson approximation (IBA) associated with u(6) and the symplectic model based on sp(6,R).
Comment on the two ‘‘new’’ classes of Bianchi type II solutions
Authors: Lorenz-Petzold Dieter
Published: July 2002

Hajj-Boutros’s claim [J. Math. Phys. 27, 1592 (1986)] that two new classes of Bianchi type II solutions can be generated from Lorenz’s solution [Phys. Lett. A 79, 19 (1980)] is shown to be wrong.
Quantum noncompact σ models
Authors: van Holten J. W.
Published: July 2002

Two possible quantization procedures for field theories with noncompact symmetry are discussed: one in a positive-definite Hilbert space with negative-energy states and one in a Hilbert space with indefinite metric and positive-definite energy. The physical interpretation of these alternative procedures is explained in terms of the realization of the noncompact symmetry, which is broken in one case and not in the other. It is shown that in exactly solvable quantum σ models the noncompact symmetry is broken in the vacuum. Some arguments supporting the general validity of this result are given. It is concluded that in contrast to theories like quantum electrodynamics, the noncompact σ models are to be quantized in the positive-definite Hilbert space with negative-energy modes.
Generalized Euler angles as intrinsic coordinates for nonlinear spinors
Authors: Furlan P.
Published: July 2002

Among the nonlinear spinor representations of the pseudo-orthogonal rotation groups presented in a preceding work, those ones are considered, whose carrier spaces are isomorphic with group spaces. The general element of such groups is written as a product of one-parameter subgroups, allowing, for the SO(ν,ν) and SO(ν+1,ν) cases, to write explicitly the nonlinear spinor components in terms of independent intrinsic coordinates satisfying the Cartan’s quadratic constraints.
The Korteweg–de Vries hierarchy of isospectral transformations: Towards a general explicit expression
Authors: Rosenhouse Avia , Katriel Jacob
Published: July 2002

The structure of the Korteweg–de Vries hierarchy of evolution equations, generating isospectral transformations, is elucidated by means of a study of its recurrence relations. For the mth member of the KdV hierarchy, which can be written in the form Vt=−2Am+1,x, where the Ai satisfy the recurrence relation Am+1,x=VAm,x+ 1/2  AmVx− 1/4  Am,xxx, it is shown that Am is a homogeneous polynomial in ∂iV/∂xi. A general combinatorial formula for the coefficients of all the monomials entering Am, up to a set of constants determined by means of a recurrence relation, is derived.
A geometric approach to quantum scattering with group symmetry
Authors: Wu Yihren
Published: July 2002

A geometric theory for quantum scattering when the symmetry group is semisimple is presented. This theory is seen as a generalization of the partial wave analysis. As an application of this theory, the S-matrix elements for scattering in the Pöschl–Teller potential with symmetry group SO(1,2), Coulomb potential with SO(1,3), and a perturbed Coulomb potential with SO(2,3) are calculated. The last example may be considered as a model for heavy-ion scattering.
Constructive representations of propagators for quantum systems with electromagnetic fields
Authors: Osborn T. A. , Papiez L. , Corns R.
Published: July 2002

The quantum evolution of an N-body system of particles that mutually interact through scalar fields and couple to an arbitrary external electromagnetic field is rigorously described. Both operator and kernel valued solutions to the evolution problem are found. Based upon a particular realization of the Dyson expansion, a convergent series representation of the propagator (the kernel of the Schrödinger time evolution operator) is obtained. The basic approach is to embed the quantum evolution problem in the larger class of evolution problems that result if mass is allowed to be complex. Quantum evolution with real mass is considered to be the boundary value of the complex mass evolution problem. The constructive representation of the propagator is determined for the class of analytic scalar and vector fields that are given as Fourier transforms of time-dependent scalar and vector-valued measures.
Tensor formulation of Hamilton’s equations
Authors: Lim Paul H.
Published: July 2002

Hamilton’s equations are presented in manifestly covariant form. The resulting equations of motion are solved via a covariant Hamilton–Jacobi scheme. A covariant correspondence principle is introduced, and it is employed to quantize the equations.
On the quantization of constrained generalized dynamics
Authors: Galvão Carlos A. P. , Lemos Nivaldo A.
Published: July 2002

A special class of degenerate second-order Lagrangians, those that differ from a nondegenerate first-order Lagrangian by a total time derivative (or a four-divergence) of a function of both the coordinates and velocities, is studied in detail. Using Dirac’s theory of constrained systems, it is shown that the canonical quantization starting from the second-order Lagrangian leads to the same physical results as those obtained from the nondegenerate first-order Lagrangian. Then some incorrect results and misleading arguments encountered in the literature on the subject are clarified.
Spinor fields and the GL(4,R) gauge structure in the nonsymmetric theory of gravitation
Authors: Moffat J. W.
Published: July 2002

The spinor structure associated with the local gauge group GL(4,R) of the nonsymmetric gravitation theory (NGT) is based on a spinor wave equation constructed from a vierbein, a GL(4,R) spin connection, and the infinite-dimensional irreducible representations of the universal covering group 𝒮ℒ (4,R) of the noncompact group SL(4,R). The multiplicity-free irreducible representations of 𝒮ℒ (4,R) correspond to bivalued spinorial representations of SL(4,R) that contain an infinite number of half-odd integer spin particles. By adjoining the translations T4, the extended group 𝒜=T4×GL(4,R) replaces the Poincaré group 𝒫. The properties of the mass spectrum are obtained from an infinite-component wave equation and the physical spinor field consists of an infinite sum of finite, nonunitary representations of the Lorentz group.
Moment problem formulation of the simplified ideal magnetohydrodynamics ballooning equation
Authors: Handy Carlos R. , Bessis Daniel , Williams Robert M.
Published: July 2002

A fundamentally new method for determining the eigenvalues of linear differential operators is presented. The method involves the application of moment analysis and affords a fast and precise numerical algorithm for eigenvalue computation, particularly in the intermediate and strong coupling regimes. The most remarkable feature of this approach is that it provides exponentially converging lower and upper bounds to the eigenvalues. The effectiveness of this method is demonstrated by applying it to an important magnetohydrodynamics problem recently studied by Paris, Auby, and Dagazian [J. Math. Phys. 27, 2188 (1986)]. Through the very precise lower and upper bounds obtained, this approach gives full support to their analysis.
Massless scalar fields at null and spatial infinity in the Schwarzschild space-time
Authors: Habisohn Chris X.
Published: July 2002

It is known that massless scalar, Maxwell, and linearized metric fields (in an appropriate gauge) having data of compact support will evolve to be asymptotically flat on any asymptotically flat background space-time. However, little is known about the evolution of data that is reasonably well behaved but has nontrivial falloff at spatial infinity. Is the set of such data that evolves to be asymptotically flat at null infinity in a curved asymptotically flat space-time of the same size as, and does it consist of elements with falloff rates similar to the set of such data in Minkowski space-time? Stewart and Schmidt analyzed massless scalar fields on both the Minkowski and Schwarzschild space-times. Their calculations indicated that the set of Schwarzschild data in question was much smaller than the Minkowski set. In this paper, this problem is reexamined and it is determined, contrary to the indications of Stewart and Schmidt, that the Schwarzschild set is of the same size, and its elements have falloff rates similar to the corresponding Minkowski set. This result supports the ability of the definition of asymptotic flatness to admit a large class of space-times.
Second-order equations from a second-order formalism
Authors: Hammond Richard T.
Published: July 2002

It is often assumed that Lagrangians of gravitation that are quadratic in the curvature tensor produce field equations of fourth differential order in the metric tensor from a Hilbert variational principle. It is shown here, for the Lagrangian given by R+RμνRμν, that independent variations of the metric tensor and the torsion tensor produce gravitational field equations of second, not fourth, differential order.
The metric of a space and quadratic algebras
Authors: Wene G. P.
Published: July 2002

A collection of algebras with product that preserves the metric of the underlying vector space is investigated. It is shown that this collection includes the generalized quaternions, the composition algebras, and the algebra of color. All the algebras in this collection that are either commutative or alternative are determined; the ramifications of nondegenerate metrics are detailed. If the algebra is alternative and not of dimension 1, 2, 4, or 8 over the base field, the metric must be degenerate.
On topological effects in quantum mechanics: The harmonic oscillator in the pointed plane
Authors: Doebner H. D. , Elmers H. J. , Heidenreich W. F.
Published: July 2002

Quantum mechanics of a (nonrelativistic) system S localized on a topologically nontrivial manifold ℳ as its configuration space is based on a quantization method, which, in general, reflects global properties of ℳ, i.e., some of the observables of S will ‘‘feel’’ the topology: There are topological effects and inequivalent quantizations on ℳ. Some straightforward examples are given for such effects, using Borel quantization (BQ), the pointed plane as manifold ℳ, and the energy operator with harmonic potential as observable. Two topological effects exist. There are unitarily inequivalent BQ on ℳ, which are equivalent to the usual quantization on the plane with a topological potential, which has the form of a Bohm–Aharonov potential. There are different self-adjoint extensions of the energy operator for a given BQ that in some cases are related to another kind of topological potential. These effects are discussed in detail, especially the self-adjoint extensions of the energy operator. An experimental setup to verify some of the results is suggested.
On the universality of linear Lagrangians for gravitational field
Authors: Jakubiec Andrzej , Kijowski Jerzy
Published: July 2002

For a large class of purely metric, metric–affine, and purely affine theories of gravitation with nonlinear Lagrangians, it is proved that the theory is equivalent to the standard Einstein theory of gravitation interacting with additional matter fields.
The gravitational field of a spinning pencil of light
Authors: Mitskievic Nikolai Vsevolodovich , Kumaradtya Krishnadeva K.
Published: July 2002

An exact solution of Einstein’s equations in a vacuum (outside of singularities), belonging to Kundt’s class and Petrov type N, is interpreted as the metric of a spinning pencil of light (a linear source infinitely extended in one direction and moving with the speed of light). It is shown that the gravitational fields of two parallel pencils of light do not interact with each other, i.e., the superposition of the metrics of two parallel pencils of light is an exact solution of Einstein’s equations in a vacuum.
Application of the eigenfunction method to the icosahedral group
Authors: Liu Fa , Ping Jia-Lun , Chen Jin-Quan
Published: July 2002

The group table for the icosahedral group I is constructed by using the isomorphism between the group I and a subgroup of the permutation group S12. The single-valued irreducible representations and Clebsch–Gordan (CG) coefficients of I are calculated by a computer code based on the eigenfunction method. The irreducible matrix elements for all the 60 group elements are given explicitly in the form of (m/n)1/2[exp(iφ)]p [2 cos φ]q [2 cos 2φ]r, where m, n, p, q, and r are integers and φ=2π/5. The Clebsch–Gordan coefficients of I are all real under a new phase convention for time reverse states and tabulated in the form of (m/n)1/2.
Classification of gauge groups in terms of algebraic structure of first class constraints
Authors: Sugano Reiji , Kimura Toshiei
Published: July 2002

Properties of gauge transformations for singular Lagrangians are investigated to classify types of gauge groups. A general method of the classification is proposed based on properties of structure functions of the Poisson brackets (or the commutators) of first class constraints. A remarkable result is that the algebraic structure of the gauge group is essentially determined by the first class constraints of the final step of constraint series which are required successively from the stationarity conditions of the constraints. Owing to this consequence, the classification of gauge groups is made simple and transparent. The structure and property of the gauge group can be characterized in terms of the algebraic structure functions among the final step constraints and the number of the steps of the constraints series. The formulation proposed will give a clue to find new types of gauge groups.
Kepler-type dynamical symmetries of long-range monopole interactions
Authors: Cordani B. , Fehér L. Gy. , Horváthy P. A.
Published: July 2002

A general procedure for understanding Kepler-type dynamical symmetries is presented. The main concern is the geodesic motion in Euclidean Taub–NUT space, which approximates the scattering of self-dual monopoles for long distances. Other examples include a test particle moving in the asymptotic field of a self-dual monopole and two other related metrics.
First integrals for the equations of motion for test particles with internal structure
Authors: Fuchs H.
Published: July 2002

In the framework of a general Lagrangian approach to the equations of motion for test particles with internal structure symmetry conditions and the corresponding conserved quantities are derived. As simple examples, first integrals of the equations of geodesic deviation, the Mathisson–Papapetrou and the Wong equations are rediscovered.
Area-preserving diffeomorphisms of the tetrahedron
Authors: Wolski A. , Dowker J. S.
Published: July 2002

Expressions for the curvature and structure constants of the area-preserving diffeomorphism group of the tetrahedron are derived. Finite-dimensional algebras are identified that yield the diffeomorphism Lie algebra in an infinite limit.
Pulse formation in a dissipative nonlinear system
Authors: Matsuno Y.
Published: July 2002

A nonlocal nonlinear evolution equation is proposed that describes pulse formation in a dissipative system. A novel feature of the equation is that it can be solved exactly through a linearization procedure. The solutions are constructed under appropriate initial and boundary conditions and their properties are investigated in detail. Of particular interest is pulse formation, which is caused by a balance between nonlinearity and dissipation. The asymptotic behavior of the solution for large time is then represented by a train of moving pulses with equal amplitudes. The corresponding position of each pulse is shown to be characterized by the zero of the Hermite polynomial, irrespective of initial conditions.
WKB equivalent potentials for q-deformed harmonic and anharmonic oscillators
Authors: Bonatsos Dennis , Daskaloyannis C. , Kokkotas K.
Published: July 2002

WKB equivalent potentials (WKB-EP’s) giving the same WKB spectrum as the q-deformed harmonic oscillator are determined in analytic form. For q being complex the WKB-EP resembles the Pöschl–Teller, Morse, or Woods–Saxon potentials, widely used in nuclear, hypernuclear, and molecular physics. For q real, the WKB-EP still goes to infinity at large x, while at small x it resembles a sextic oscillator. The WKB-EP’s of the anharmonic oscillators with Uq(2) or SUq(1,1) symmetries, used for the description of vibrational molecular spectra, are also calculated. They correspond to q-deformed versions of the modified Pöschl–Teller potential.
On the stability of collapse in the critical case
Authors: Laedke E. W. , Blaha R. , Spatschek K. H. , Kuznetsov E. A.
Published: July 2002

Collapsing solutions in the Cauchy problem of the nonlinear Schrödinger equation i ∂tψ + ∇2ψ +‖ ψ‖pψ = 0 (x∈ℛd) are considered in the so-called critical case pd=4, where d is the spatial dimension. A stability theorem for radial collapse is presented which proves that the formation of the singularity remains ‘‘close’’ to the self-similar collapsing solution with a spatial profile given by the ground state solitary wave, provided the energy H{ψ}<0. An analogous result is given for the scalar Zakharov equations and a generalized Korteweg–de Vries equation ∂tu + (p + 1)up ∂xu + ∇2 ∂xu = 0.
Gravitational interaction of plane gravitational waves and matter shells
Authors: Wang Anzhong
Published: July 2002

The Weyl, Ricci scalars and the Bianchi identities are explicitly given in terms of distributions. Using the obtained Bianchi identities, the gravitational interaction of colliding shock or impulsive gravitational plane waves or matter shells is studied. It is found that, when two impulsive shells of null dust collide, they necessarily produce a tail. The tail could be a ‘‘Coulomb-like’’ gravitational field, or a matter current, or a mixture of both.
Hilbert superspace
Authors: Nagamachi Shigeaki , Kobayashi Yuji
Published: July 2002

A theory of Hilbert superspace over an infinite dimensional Grassmann algebra Λ is given. Axioms of Hilbert superspace are given and it is proven that a Hilbert superspace is isomorphic to H⊗Λ for some Hilbert space H. A natural topology on it called ε topology is defined and continuous Λ-linear operators, especially unitary operators are studied. A Hilbert subsuperspace is defined and it is proven that its orthogonal complement is a Hilbert subsuperspace.
The q-boson realization of parametrized cyclic representations of quantum algebras at q
p = 1
Authors: Fu Hong-Chen , Ge Mo-Lin
Published: July 2002

A parametrized cyclic representation of the q-deformed Heisenberg–Weyl algebras is constructed. The cyclic representations of any quantum algebras are studied in terms of their q-boson realizations. A general method to construct the q-boson realizations of quantum algebras from their Verma representations is proposed. Two explicit examples sl(2)q and sl(3)q are studied in detail. Some cyclic representations of quantum algebras sl(n)q and UqCn are presented by using their q-boson realizations.
On the semiclassical localization of the quantum probability
Authors: Duclos P. , Hogreve H.
Published: July 2002

The localization behavior of one-dimensional quantum systems for ℏ→0 is investigated by semiclassical methods. In particular the localization of the quantum probability around turning points of arbitrary even order associated to classical hyperbolic orbits is considered and a relation of the localization speed in ℏ with the classical motion is established. Our analysis is based on local norm comparisons of solutions to Schrödinger type equations; it relies mainly on a combination of scaling and asymptotic arguments and thus evades the use of special functions. Applications of the results to separable multidimensional Schrödinger equations are indicated by a brief discussion of the one-electron diatomic molecular ion.
A Lie algebraic study of some Schrödinger equations
Authors: Burrows B. L. , Cohen M. , Feldmann Tova
Published: July 2002

Bound-state solutions of several different Schrödinger equations are calculated rather efficiently using the methods of Lie algebra. In all cases considered, either of the algebras SO(3) or SO(2,1) provides a suitable framework, and there is no advantage to either choice. However, the choice of an appropriate realization of the generators is of greater significance, leading to a particularly simple solution whenever the Hamiltonian can be expressed as a linear function of the generators. In certain cases, the Hamiltonian can be expressed only as a bilinear function of the generators and only part of the bound-state spectrum can be calculated analytically to yield a finite set of so-called quasiexact solutions. These may be interrelated by means of suitable ladder operators and it is possible, though by no means necessary, to adopt a particular finite-dimensional representation of the underlying algebra. The present work emphasizes the role of similarity transformations, and the connection between the Lie method and other (generally variational) procedures which lead to the diagonalization of very large (in principle infinite-dimensional) matrices.
Topological censorship and the topology of black holes
Authors: Browdy Steven F. , Galloway Gregory J.
Published: July 2002

A new result on topological censorship is presented. Roughly speaking, the result shows that, under suitable conditions, the topology of space outside the event horizon must eventually be simple. In particular, cross sections of the event horizon must eventually be spherical. This latter consequence of the main result generalizes in many respects the classical theorem of Hawking that black hole boundaries in stationary black hole space–times are spherical.
Symmetries, parametrization, and group structure of transfer matrices in quantum scattering theory
Authors: Pereyra Pedro
Published: July 2002

Useful and new representations of the transfer matrices are explicitly obtained for systems with and without time reversal or spin rotation symmetries. In the symplectic, orthogonal, and unitary cases, the whole group of transfer matrices contains a compact and a noncompact subgroup, and the generalized Bargman’s polar parametrization is obtained in a straightforward way. It is shown that the transfer matrices used by Luttinger, to prove the Saxon–Hutner conjecture on the forbidden energy levels, belong to the noncompact Sp(2,𝒞) subgroup.
Integrable dynamics of a discrete curve and the Ablowitz–Ladik hierarchy
Authors: Doliwa Adam , Santini Paolo Maria
Published: July 2002

It is shown that the following elementary geometric properties of the motion of a discrete (i.e., piecewise linear) curve select the integrable dynamics of the Ablowitz–Ladik hierarchy of evolution equations: (i) the set of points describing the discrete curve lie in the sphere S3; (ii) the distance between any two subsequent points does not vary in time; (iii) the equations of the dynamics do not depend explicitly on the radius of the sphere. These results generalize to a discrete context the previous work on continuous curves [A. Doliwa and P. M. Santini, Phys. Lett. A 185, 373 (1994)].
q-deformed Chern characters for quantum groups SUq(N)
Authors: Hou Bo-Yu , Hou Bo-Yuan , Ma Zhong-Qi
Published: July 2002

In this paper, we introduce an N×N matrix εab̄ in the quantum groups SUq(N) to transform the conjugate representation into the standard form so that we are able to compute the explicit forms of the important quantities in the bicovariant differential calculus on SUq(N), such as the q-deformed structure constant CKIJ and the q-deformed transposition operator Λ. From the q-gauge covariant condition we study the generalized q-deformed Killing form and construct the mth q-deformed Chern class Pm for the quantum groups SUq(N). In terms of the q-deformed homotopy operator we are able to compute the q-deformed Chern–Simons Q2m−1 by the condition dQ2m−1=Pm. Furthermore, the q-deformed cocycle hierarchy, the q-deformed gauge covariant Lagrangian, and the q-deformed Yang–Mills equation are derived.
Dynamics of relativistic particles with Lagrangians dependent on acceleration
Authors: Nesterenko V. V. , Feoli A. , Scarpetta G.
Published: July 2002

Models of relativistic particles with Lagrangians ℒ(k1), depending on the curvature of the worldline k1, are considered. By making use of the Frenet basis, the equations of motion are reformulated in terms of the principal curvatures of the worldline. It is shown that for arbitrary Lagrangian function ℒ(k1) these equations are completely integrable, i.e., the principal curvatures are defined by integrals. The constants of integration are the particle mass and its spin. The developed method is applied to the study of a model of a relativistic particle with maximal proper acceleration, whose Lagrangian is uniquely determined by a modified form of the invariant relativistic interval. This model gives us an example of a consistent relativistic dynamics obeying the principle of a superiorly limited value of the acceleration, advanced recently.
Localization in single Landau bands
Authors: Dorlas T. C. , Macris N. , Pulé J. V.
Published: July 2002

We consider a single-band approximation to the random Schrödinger operator in an external magnetic field. The random potential is taken to be constant on unit squares and i.i.d. on each square with a bounded distribution. We prove that the eigenstates corresponding to energies at the edges of the Landau band are localized. This is an important ingredient in the theory of the Quantum Hall Effect.
Radial Coulomb and oscillator systems in arbitrary dimensions
Authors: Kostelecký V. Alan , Russell Neil
Published: July 2002

A mapping is obtained relating analytical radial Coulomb systems in any dimension greater than one to analytical radial oscillators in any dimension. This mapping, involving supersymmetry-based quantum-defect theory, is possible for dimensions unavailable to conventional mappings. Among the special cases is an injection from bound states of the three-dimensional radial Coulomb system into a three-dimensional radial isotropic oscillator where one of the two systems has an analytical quantum defect. The issue of mapping the continuum states is briefly considered.
Bounds on number of cusps due to point mass gravitational lenses
Authors: Petters A. O. , Witt H. J.
Published: July 2002

Generic caustics in gravitational lensing occur locally either as folds or cusps. This paper rigorously proves that the total number of cusps, Ncusps, due to g point masses on a single plane having non-normalized external shear γ≳0 and continuous matter with constant density σc, is bounded as follows: 0≤Ncusps≤12g2. For vanishing shear γ=0 we obtain the result 0≤Ncusps≤12g(g−1). Consequences of these bounds for the global geometry of caustics are discussed. It is also shown that if γ≥0 and σc is sufficiently large, then all cusps can be eliminated, that is, Ncusps=0. The paper also includes equations for calculating all the bi-caustics (i.e., curves yielding the positions of cusps during a one-parameter evolution) of a single point-mass lens with continuous matter and shear. The methods of the paper are based on a new approach to point-mass gravitational lensing using complex quantities and the theory of resultants.
Response to the Comment by G. Emch on projective group representations in quaternionic Hilbert space
Authors: Adler Stephen L.
Published: July 2002

We discuss the differing definitions of complex and quaternionic projective group representations employed by us and by Emch. The definition of Emch (termed here a strong projective representation) is too restrictive to accommodate quaternionic Hilbert space embeddings of complex projective representations. Our definition (termed here a weak projective representation) encompasses such embeddings, and leads to a detailed theory of quaternionic, as well as complex, projective group representations.
A calculus for SU(3) leading to an algebraic formula for the Clebsch–Gordan coefficients
Authors: Prakash J. S. , Sharatchandra H. S.
Published: July 2002

We develop a simple computational tool for SU(3) analogous to Bargmann’s calculus for SU(2). Crucial new inputs are (i) explicit representation of the Gelfand–Zetlin basis in terms of polynomials in four variables and positive or negative integral powers of a fifth variable, (ii) an auxiliary Gaussian measure with respect to which the Gelfand–Zetlin states are orthogonal but not normalized, (iii) simple generating functions for generating all basis states and also all invariants. As an illustration of our techniques, an algebraic formula for the Clebsch–Gordan coefficients is obtained for the first time. This involves only Gaussian integrations. Thus SU(3) is made as accessible for computations as SU(2) is.
Krein duality for compact quantum groups
Authors: Wang Shuzhou
Published: July 2002

We define noncommutative Krein algebras and extend the classical Krein duality for compact groups to compact quantum groups.
An inverse scattering transform for the MKdV equation with non-vanishing boundary value
Authors: Huang Nian-Ning , Chen Zong-Yun , Chen Xiang-Jun
Published: July 2002

The MKdV equation of normal dispersion with non-vanishing boundary value is solved by the inverse scattering transform method. An affine parameter is introduced to avoid double-valued functions of the usual spectral parameter. In terms of it the inverse scattering transform is performed and the inverse scattering equation of Zakharov–Shabat form as well as of Marchenko form is derived. Dark multi-soliton solutions are found formally by means of the Binet–Cauchy formula. The asymptotic behaviors in the limits of |t|→∞ are derived as expected.
Construction of a complete set of states in relativistic scattering theory
Authors: Pelc Oskar , Horwitz L. P.
Published: July 2002

The space of physical states in relativistic scattering theory is constructed, using a rigorous version of the Dirac formalism, where the Hilbert space structure is extended to a Gel’fand triple. This extension enables the construction of “a complete set of states,” the basic concept of the original Dirac formalism, also in the cases of unbounded operators and continuous spectra. We construct explicitly the Gel’fand triple and a complete set of “plane waves”—momentum eigenstates—using the group of space–time symmetries. This construction is used (in a separate article) to prove a generalization of the Coleman–Mandula theorem to higher dimension.
Algebraic expressions for irreducible bases of icosahedral group
Authors: Chen Jin-Quan , Ping Jia-Lun
Published: July 2002

The recently discovered point-group symmetrized boson representation (SBR) is constructed for the icosahedral group Ih. With the aid of the SBR, succinct algebraic expressions for the irreducible bases and irreducible matrices have been found. Irreducible bases for non-regular representations can be found easily from those of the regular representation without projections. Explicit expressions of the irreducible bases are given for the molecule B12H12 for several important cases. It is shown that, far from being “chaotic” in structure, the 120 irreducible matrices of Ih have a high degree of symmetry in that the 14 400 entries can be reproduced from a few dozen entries according to three rules.
Quaternionic representations of magnetic groups
Authors: Scolarici G. , Solombrino L.
Published: July 2002

We study magnetic groups, which contain a time-inversion operator besides spatial symmetries, in the framework of quaternionic group representation theory. We obtain a classification of these groups, depending on the reducibility of their spatial part, and then we cross it with the generalized Frobenius–Schur classification that has been obtained by ourselves elsewhere. Ten distinct cases arise, but only five of them apply to factorizable groups (i.e., groups in which the time-inversion operator appears alone and not multiplied by a spatial symmetry). We supply examples for these five cases and determine which of them apply to bosonic or fermionic systems, respectively. Finally, we discuss the degeneracy of energy levels in the presence of a time-reversal symmetry (Kramers degeneracy) in quaternionic quantum mechanics, obtaining right values in all cases.
Accuracy of the Hartree–Fock approximation for the Hubbard model
Authors: Bach Volker , Poelchau Jurij
Published: July 2002

A lower bound 0⩾(Egs(n)−Ehf(n))/|Λ|⩾−const[n2/3U4/3(|ln U|+1) +Un1/2|Λ|−1/2d(|ln(|Λ|−1/2)|1)] to the difference of the ground state and the Hartree–Fock energy of the Hubbard model is derived. Here |Λ| is the lattice size, U is the coupling parameter, and n is the electron density per site. This estimate holds for all dimensions d⩾2 and all densities. Thus the Hartree–Fock approximation becomes exact (even beyond terms of order U) for small U and large |Λ|.
Signs and approximate magnitudes of Lyapunov exponents in continuous time dynamical systems
Authors: Birol İnanç , Hacinliyan Avadis
Published: July 2002

Methods for algebraically determining the signs and the magnitudes of Lyapunov exponents of a given dynamical system are studied. The existence of zero Lyapunov exponents for the Toda, Hénon–Heiles, and Rössler systems are shown. The approximate Lyapunov spectra of Lorenz and Rössler systems are computed.
A new boson calculus for SU(3)
Authors: Prakash J. S. , Sharatchandra H. S.
Published: July 2002

It is shown that the Gelfand-Weyl pattern for SU(3) can be uniquely expressed in terms of four non-negative and one positive or negative free integers. This provides an optimal, albeit non-linear, boson calculus for SU(3) in terms of four harmonic oscillators and one planar rotor. (By optimal it is meant that every irreducible representation is obtained once and once only.) Our techniques can be generalized to other groups.
Supersymmetry and supercoherent states of a nonrelativistic free particle
Authors: Samsonov Boris F.
Published: July 2002

Coordinate atypical representation of the orthosymplectic superalgebra osp (2/2) in a Hilbert superspace of square integrable functions constructed in a special way is given. The quantum nonrelativistic free particle Hamiltonian is an element of this superalgebra which turns out to be a dynamical superalgebra for this system. The supercoherent states, defined by means of a supergroup displacement operator, are explicitly constructed. These are the coordinate representation of the known atypical abstract super group OSp (2/2) coherent states. We interpret obtained results from the classical mechanics viewpoint as a model of classical particle which is immovable in the even sector of the phase superspace and is in rectilinear movement (in the appropriate coordinate system) in its odd sector.
Excitations in one dimension: A geometrical view of the transfer matrix method
Authors: Destainville N. , Sadoc J. F.
Published: July 2002

The transfer matrix formalism is a useful tool for the study of excitations in one-dimensional chains. The geometrical approach presented here maps the dynamic of transfer matrix products onto a dynamic on the Poincaré disc, a suitable model for the hyperbolic plane. This mapping uses a fibration of the three dimensional manifold on which the transfer matrices lie: the transfer matrix dynamic is mapped onto a polygonal trajectory on the base of this fibration. The link between the behavior of these trajectories and the spectrum of the system under study is discussed. We particularly focus on approximants of quasiperiodic systems. We also give a geometrical construction of the so-called trace-map for quasiperiodic systems.
Gamow-Jordan vectors and non-reducible density operators from higher-order S-matrix poles
Authors: Bohm A. , Loewe Mark , Maxson S. , Patuleanu P. , Püntmann C. , Gadella M.
Published: July 2002

In analogy to Gamow vectors that are obtained from first-order resonance poles of the S-matrix, one can also define higher-order Gamow vectors which are derived from higher-order poles of the S-matrix. An S-matrix pole of r-th order at zR=ER−iΓ/2 leads to r generalized eigenvectors of order k=0,1,…,r−1, which are also Jordan vectors of degree (k+1) with generalized eigenvalue (ER−iΓ/2). The Gamow-Jordan vectors are elements of a generalized complex eigenvector expansion, whose form suggests the definition of a state operator (density matrix) for the microphysical decaying state of this higher-order pole. This microphysical state is a mixture of non-reducible components. In spite of the fact that the k-th order Gamow-Jordan vectors has the polynomial time-dependence which one always associates with higher-order poles, the microphysical state obeys a purely exponential decay law.
On the energy spectrum of the hydrogen atom in a photon field. I
Authors: Prosser Reese T.
Published: July 2002

In 1930 Weisskopf and Wigner gave an account, based on the Maxwell–Schrödinger equations, of the natural spectral line broadening of radiation emitted by a hydrogen atom. Their calculations were based on an approximation involving certain single-photon transitions in the perturbation series for the solutions of these equations. In Part I of this series of papers the exact expressions for both the line shift and the line broadening are obtained from the Maxwell–Dirac equations in such a way that the Weisskopf–Wigner results appear as a second order approximation. The Maxwell–Dirac Hamiltonian for the coupled fields is first shown to admit a complex analytic dilation in the energy variables. The Fredholm–Born series for the resolvent is shown to converge uniformly when certain high-energy cutoff factors are included in the interaction and the photons are given a small mass. The series is then rearranged to show that the spectrum of the modified dilated Hamiltonian, which consists of a complete set of complex eigenvalues, thresholds, and branch cuts, is only a slight perturbation of the known spectrum of the dilated Hamiltonian for the uncoupled fields. The real part of the shift of each complex eigenvalue then accounts for the spectral line shift, and the complex part accounts for the associated line broadening. Finally, the implications for the scattering matrix and the various phenomena of resonance scattering are discussed. In Part II of this series these results are shown to remain valid when the cutoff factors and the photon mass are removed.
The Edmonds asymptotic formulas for the 3j and 6j symbols
Authors: Flude James P. M.
Published: July 2002

The purpose of this paper is to provide definitions for, and proofs of, the asymptotic formulas given by Edmonds, which relate the 3j and 6j symbols to rotation matrices.
Algebraic expressions of the Clebsch-Gordon coefficients of the point group T†
Authors: Chen Jin-Quan , Fan Peng-Dong
Published: July 2002

A method for finding algebraic expressions of the Clebsch-Gordan (CG) coefficients of point groups is proposed and applied to the tetrahedral group. It is shown that in constructing the CG coefficients the irreducible symmetry operator (ISO) of a double group G† can be replaced by an effective ISO which is much simpler than the usual ISO. The effective ISO for the group chain T†⊃C3† is Pμ,μ̄(λ)=δμμ̄+3dμμ̄(λ)(C2z)*C2z, where d(λ)(C2z) is the matrix of C2z in the irrep λ of T†. With this effective ISO and the algebraic expression of d(λ)(C2z), the algebraic expressions are derived for the real CG coefficients of T† in the group chain T†⊃C3†. The algebraic expressions for the complex (real) CG coefficients of the group chain T†⊃D2†⊃C2† (T†⊃C2†) have also been obtained.
An existence theory for relativistic brachistochrones in stationary space–times
Authors: Giannoni Fabio , Piccione Paolo
Published: July 2002

We use a variational principle proven by Giannoni et al. [J. Math. Phys. 38, 6367–6381 (1997)] for relativistic brachistochrones and techniques of global analysis to develop an existence theory for brachistochrones in stationary space–times. We also develop a Ljusternik–Schnirelman theory for brachistochrones, obtaining multiplicity results depending on the topology of the space–time.
Complete sets of Bloch and Wannier functions composed of oscillator eigenfunctions
Authors: Zeiner P. , Dirl R. , Davies B. L.
Published: July 2002

We discuss Bloch and Wannier functions related to oscillator eigenfunctions. In particular, we construct complete sets of mutually orthogonal Bloch and Wannier functions. We show that they can be expressed in several ways in terms of theta functions and their derivatives. We also analyze their localization properties and discuss expectation values for specifically chosen Hamiltonians.
An estimate of the ground state energy of the fractional quantum Hall effect
Authors: Xia Jingbo
Published: July 2002

Suppose that there are N electrons in a disk of radius R with a perpendicular magnetic field. We give an estimate for the ground state energy of such a system in the case N≈νR2.
New approach to semiclassical analysis in mechanics
Authors: Alonso M. A. , Forbes G. W.
Published: July 2002

A new method is proposed for constructing approximate solutions to the Schrödinger equation. In place of the wave function, its Gaussian-windowed Fourier transform is used as the fundamental entity. This allows an intuitively attractive connection to be made with a family of classical trajectories and, at all times, the wave function is inferred from the present state of these trajectories. The fact that the connection between the wave function and the classical trajectories is consistently constructed in phase space allows this method to be free of the limitations of other methods.
Differential equations for scaling relation in N=2 supersymmetric SU(2) Yang–Mills theory coupled with massive hypermultiplet
Authors: Ohta Yűji
Published: July 2002

Differential equations for the scaling relation of prepotential in N=2 supersymmetric SU(2) Yang–Mills theory coupled with massive matter hypermultiplet are proposed and are explicitly demonstrated in one flavor (Nf=1) theory. By applying Whitham dynamics, the first-order derivative of the prepotential over the T0 variable corresponding to the mass of the hypermultiplet, which has a line integral representation, is found to satisfy a differential equation. As a result, the closed form of this derivative can be obtained by solving this equation. In this way, the scaling relation of massive prepotential is established. Furthermore, as an application of another differential equation for the massive scaling relation, the massive prepotential in a strong coupling region is derived.
Asymptotically Schwarzschild space–times
Authors: Gen Uchida , Shiromizu Tetsuya
Published: July 2002

It is shown that if an asymptotically flat space–time is asymptotically stationary, in the sense that Lξgab vanishes at the rate ∼t−3 for asymptotically timelike vector field ξa, and the energy–momentum tensor vanishes at the rate ∼t−4, then the space–time is an asymptotically Schwarzschild space–time. This gives a new aspect of the uniqueness theorem of a black hole.
Generalized Kronecker and permanent deltas, their spinor and tensor equivalents and applications
Authors: Agacy R. L.
Published: July 2002

The aim of this paper is fourfold: (i) to introduce a generalized permanent delta on an equal footing with the generalized Kronecker delta, to use for the symmetries of any tensor or spinor, (ii) to cite an ancillary reference source of comprehensive tensorial and spinorial combinatorial formulas for both, (iii) to table spinor equivalents of these individual tensors and give examples of their usage, and (iv) to tabulate the tensor equivalents of various useful combinations of their spinor forms.
Waves and particles in Kaluza–Klein theory
Authors: Sajko W. N. , Wesson P. S. , Liu H.
Published: July 2002

We examine three overlapping problems in the application of five-dimensional (5D) manifolds to physics. First, we linearize the 5D theory along the lines of the four-dimensional (4D) theory, using the harmonic gauge condition. The resulting wave equations have sources, and can in principle describe gravitons and scalar particles with finite masses, but the natural choice of gauge parameters makes both massless. Second, we generalize the 5D metric by including separate conformal factors on its 4D and extra parts. Then the 5D harmonic gauge gives back the 4D harmonic gauge of gravitational waves and the Lorentz gauge of electromagnetic waves, but both particles are massless. Third, we again take a conformally rescaled metric, but rewrite the field equations in a novel manner. This allows us to interpret the finite masses of ordinary particles in terms of the wavelengths associated with 4D spaces embedded in a 5D space.
Comment on “Generalized W∞ symmetry algebra of the conditionally integrable nonlinear evolution equation” [J. Math. Phys. 36, 3492 (1995)]
Authors: Ma Wen-Xiu
Published: July 2002

Two remarks on an inverse operator of a differential operator ∂x and on symmetries of two kinds of differential equations Δ(u)=0 and ∂xΔ(u)=0 are pointed out and then a few of remarks about the paper “Generalized W∞ symmetry algebra of the conditionally integrable nonlinear evolution equation” [Y. Lou and J. P. Weng, J. Math. Phys. 36, 3492 (1995)] are presented. It is in particular shown that if we consider ∂x−1 to be a linear and right inverse operator of ∂x, the vector fields σnu(f ) proposed in the above paper are not certain to be symmetries of the Jimbo–Miwa–Kadomtsev–Petviashvili system under consideration. Moreover the commutators of a generalized W∞ algebra established among these vector fields do not always hold. Therefore the vector fields σnu(f ) do not provide an example of application of the formal series ansatz in the above paper.
Magnetic monopole in the Feynman’s derivation of Maxwell equations
Authors: Bérard A. , Grandati Y. , Mohrbach H.
Published: July 2002

In 1992, Dyson published Feynman’s proof of the homogeneous Maxwell equations assuming only the Newton’s law of motion and the commutation relations between position and velocity for a nonrelativistic particle. Recently Tanimura gave a generalization of this proof in a relativistic context. Using the Hodge duality we extend his approach in order to derive the two groups of Maxwell equations with a magnetic monopole in flat and curved spaces.