bluuebunny/crossref_metadata_2025 huggingface dataset:
Value preserving quantum measurements: Impossibility theorems and lower bounds for the distortion
Authors: Ghirardi G. C. , Rimini A. , Weber T.
Published: March 2003

Extending previous works on the subject we consider the problem of the limitations to ideal quantum measurements arising from the presence of additive conservation laws and we discuss impossibility theorems and derive lower bounds for the deviations from the ideal schemes, with particular reference to the distorting case.
Derivation of the generalized Langevin equation by a method of recurrence relations
Authors: Lee M. Howard
Published: March 2003

The generalized Langevin equation was first derived by Mori using the Gram-Schmidt orthogonalization process. This equation can also be derived by a method of recurrence relations. For a physical space commonly used in statistical mechanics, the recurrence relations are simple and they lead directly to the Langevin equation. The Langevin equation is shown to be composed of one homogeneous and one inhomogeneous equation.
Solution of the Dirac equation for the general even power potential with application to particle spectroscopy
Authors: Sharma L. K. , Mehta Sheilly , Iyer V. P.
Published: March 2003

High-energy asymptotic expansions of eigenenergies, wavefunctions, and Regge trajectories of the generalized even power potential V(r)=−g2∑∞j=0N2 jr2j in the Dirac equation are obtained. These general expansions are then used to obtain eigenenergy expansions and Regge trajectories for the anharmonic oscillator and Gauss potentials. Finally, the relevance of these investigations to the spectroscopy of heavy quark composites is discussed. In particular, we study the charmonium spectroscopy for an harmonic oscillator.
On the hydrodynamic self-similar cosmological models
Authors: Chi L. K.
Published: March 2003

The hydrodynamic self-similar cosmological models are considered. A new solution is presented. Of all the self-similar solutions, only the Newtonian analogy of the Einstein–deSitter model represents an expanding universe.
On the controllability of quantum-mechanical systems
Authors: Huang Garng M. , Tarn T. J. , Clark John W.
Published: March 2003

The systems-theoretic concept of controllability is elaborated for quantum-mechanical systems, sufficient conditions being sought under which the state vector ψ can be guided in time to a chosen point in the Hilbert space ℋ of the system. The Schrödinger equation for a quantum object influenced by adjustable external fields provides a state-evolution equation which is linear in ψ and linear in the external controls (thus a bilinear control system). For such systems the existence of a dense analytic domain 𝒟ω in the sense of Nelson, together with the assumption that the Lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension, permits the adaptation of the geometric approach developed for finite-dimensional bilinear and nonlinear control systems. Conditions are derived for global controllability on the intersection of 𝒟ω with a suitably defined finite-dimensional submanifold of the unit sphere Sℋ in ℋ. Several soluble examples are presented to illuminate the general theoretical results.
Convergence of the T-matrix approach in scattering theory. II
Authors: Kristensson G. , Ramm A. G. , Ström S.
Published: March 2003

Convergence of the T-matrix scheme is proved under more general assumptions than in Ramm [J. Math. Phys. 23, 1123–5 (1982)] and for more general boundary conditions. Stability of the numerical scheme towards small perturbations of data and convergence of the expansion coefficients are established. Dependence of the rate of convergence on the choice of basis functions is discussed. Dependence of the quality of expansions in various spherical waves on the shape of the obstacle is discussed.
Solutions of Einstein–Yang–Mills equations with plane symmetry
Authors: Mondaini R. P. , Santos N. O.
Published: March 2003

The Einstein–Yang–Mills system is solved with the assumption of plane symmetry. We present a class of abelian and a class of SO(3) nonabelian solutions of the Yang–Mills equations.
Inverse scattering variables of the KdV equation from the point of view of Galilean mechanics
Authors: Alonso L. Martínez
Published: March 2003

The Galilean invariance of the Korteweg–de Vries equation is applied in order to characterize the structure of degrees of freedom displayed by the inverse scattering transform method. It is found that the dynamical systems associated with the discrete scattering data variables admit a description in terms of mass, position, and momentum variables similar to the systems of free Galilean particles. On the other hand, the radiation component of the KdV field associated with the continuous part of the set of scattering data turns out to be described by a new field evolving according to a linear partial differential equation.
Casimir elements of ε Lie algebras
Authors: Scheunert M.
Published: March 2003

The classical framework for investigating the Casimir elements of a Lie algebra is generalized to the case of an ε Lie algebra L. We construct the standard L-module isomorphism of the ε-symmetric algebra of L onto its enveloping algebra, and we introduce the Harish–Chandra homomorphism. In case the generators of L can be written in a canonical two-index form, we construct the associated standard sequence of Casimir elements and derive a formula for their eigenvalues in an arbitrary highest weight module.
Eigenvalues of Casimir operators for the general linear, the special linear, and the orthosymplectic Lie superalgebras
Authors: Scheunert M.
Published: March 2003

The generators of the algebras under consideration can be written in a canonical two-index form and hence the associated standard sequence of Casimir elements can be constructed. Following the classical approach by Perelomov and Popov, we obtain the eigenvalues of these Casimir elements in an arbitrary highest weight module by calculating the corresponding generating functions.
Anisotropic cosmological model with viscous fluid
Authors: Banerjee A. , Santos N. O.
Published: March 2003

We present an exactly solvable Bianchi type I cosmological model with a viscous fluid. We show that the role of viscosity is more important in the initial epochs of the universe and, in that same period, pressure is more important than fluid density.
On Z
n graded Baker–Campbell–Hausdorff formulas and coset space parametrizations
Authors: Parker C. N. , Winter B. D.
Published: March 2003

We present a generalization of the BCH formula, exp Δ exp Γ=exp(Δ+Γ+ (1/2)[Δ,Γ]+⋅⋅⋅), involving n factors on the right, linear in Δ and to all orders in Γ. The result is applicable to coset geometries involving Zn graded algebras and superalgebras. In the Z2 case expressions are obtained for group transformations in the spaces Sn−1=O(n)/O(n−1), CPn=SU(n+1)/U(n), OSp(1/n)/Sp(n), and SU(n/1)/U(n).
A duality consistent phase convention for complex conjugation in SUn
Authors: Sullivan John J.
Published: March 2003

We have argued that a simple phase convention for all involutive mappings can be chosen so the algebra of inner product couplings in SUn is identical to that for outer products in SL. Pure real phases with a single ±1 entry in any row or column can represent the permutation matrix for transposition in order of the two component irreducible representations (irreps) in a binary coupling, the matrix for association in SL with respect to the alternating group AL, the Derome–Sharp matrix, and the 1jm factor. The latter two are required by complex conjugation in SUn. In our previous work, we have proposed specific prescriptions for assigning the phase under transposition and association and have shown them to be consistent with duality. In this work, we propose a duality consistent prescription for assigning the phase of the Derome–Sharp matrix and thus of the 1jm factor which is related to it by association.
Eigenvalues of Casimir operators for the general linear and orthosymplectic Lie superalgebras
Authors: Bincer Adam M.
Published: March 2003

A Racah basis is introduced for the generators of these matrix superalgebras and explicit formulas are derived for eigenvalues of Casimir operators in terms of the components of the highest weight. The result contains, as special cases, the corresponding expressions for the general linear, orthogonal, and symplectic Lie algebras.
New representations for the spherical tensor gradient and the spherical delta function
Authors: Weniger E. Joachim , Steinborn E. Otto
Published: March 2003

In this article, we analyze representations for the product 𝒴m1l1(∇)Fm2l2(r) with 𝒴ml(∇) specifying a solid harmonic whose argument is the nabla operator ∂/∂r instead of the vector r. Since both 𝒴m1l1(∇) and Fm2l2(r) are irreducible spherical tensors, we can use angular momentum algebra for evaluating the product. Accordingly, the problem of finding a representation for the product is reduced to the determination of the radial functions generated by the product. Analytical expressions for these radial functions are derived by direct differentiation and with the help of Fourier transforms. Closely related to the spherical tensor gradient 𝒴ml(∇) is the spherical delta function δml(r). We derive new representations for δml by considering convolution integrals involving B functions. These functions are closely related to the modified Bessel functions and also to the Yukawa potential e−αr/r. We show that the definition of the B functions can be extended to include a large class of derivatives of the delta functions, where the spherical delta function is just a special case.
A useful formula for evaluating commutators
Authors: Gallatin G. M.
Published: March 2003

We present the derivation of a useful formula for evaluating commutators of the form [A, f (B)] and [ f (A),B], where the nested commutators [A,[A,[⋅⋅⋅[A[A,B]]⋅⋅⋅]]] and [[[⋅⋅⋅[[A,B],B]⋅⋅⋅],B],B] do not vanish in general. The use of this formula is illustrated by a simple example.
Realization of Gaussian random fields
Authors: Krée Paul
Published: March 2003

The representation of stationary Gaussian processes in terms of filtered Gaussian white noise is studied. Known results are extended from the finite-dimensional case to the dimension-free case; hence, in particular, to Gaussian random fields. In particular, the following result is proved for usual Gaussian processes: Physical realizability is equivalent to realizability.
Solutions of Einstein’s equations involving arbitrary functions
Authors: Szekeres Peter
Published: March 2003

Necessary and sufficient conditions are derived for a solution of Einstein’s vacuum equations to depend on an arbitrary function of some scalar function φ. Unlike the case of the scalar wave equation the constant surfaces of the function φ need not be null. This apparent anomaly is discussed.
The Cauchy problem for the R+R2 theories of gravity without torsion
Authors: Teyssandier P. , Tourrenc Ph.
Published: March 2003

The exterior Cauchy problem is discussed for the fourth-order theories of gravity derived from the Lagrangian densities ℒ=(−g)1/2 (R+ (1/2)aR2+bRμν Rμν) −κℒm. When b≠0, the Cauchy problem can be solved by the standard method already used in general relativity. When b=0, the problem cannot be formulated as in the case where b≠0, since the corresponding fourth-order theory is shown to be equivalent to a second-order scalar–tensor theory. This scalar–tensor theory is proved to coincide with one of the models of gravity proposed by O’Hanlon in order to present a covariant version of the massive dilaton theory suggested by Fujii. This result is generalized: The models of O’Hanlon are shown to be indistinguishable from the fourth-order theories derived from the Lagrangian densities ℒ=(−g)1/2 F(R)−κℒm, where F is any real function such that F″(R) does not identically vanish.
A probabilistic rejection test for multivariable sensitivity analysis
Authors: Ronen Y. , Dubi A.
Published: March 2003

A probabilistic rejection test for multivariable sensitivity analysis is presented. The test is applied by randomly changing all the assumed unimportant (those having low sensitivity values) input parameters simultaneously and calculating the appropriate response. It is shown that by repeating this procedure N times, where N is much smaller than the number of input parameters, it is possible to assign a probability limit to the assumption that a high sensitivity parameter exists. The application of the test is demonstrated in a nuclear waste disposal problem.
A Lagrangian of Bargmann–Wigner equations for massive particles of spin 2
Authors: Lorente M. , Rodriguez M. A.
Published: March 2003

Following the ideas of Guralnik and Kibble and those of Larsen and Repko, we introduce a general method to calculate the first-order Lagrangian of Bargmann–Wigner equations (BWE) of arbitrary spin, and make explicit calculations in case of spin 2. Finally, some considerations on the motivation of this method and on the invariance of the Lagrangian under the symmetric group and the general Lorentz group are discussed.
Cross sections with polarized spin-1/2 particles in terms of helicity amplitudes
Authors: Anders Till B. , Jachmann Wolfgang
Published: March 2003

The derivation of cross sections for collisions with polarized particles of spin 1/2 can be simplified considerably if the scattering amplitude is calculated explicitly before the transition probability is obtained by a simple squaring. The states of the polarized particles are represented as superpositions of states with definite helicity. The coefficients of the superposition relate directly to the strength of the transversal and longitudinal polarization. The helicity amplitudes are products of helicity currents for which detailed formulas have been elaborated. Two-component spinors have been used. All the contractions of vector indices are done with a new set of general formulas. The resulting cross sections show terms with separate factors due to the polarization, the energy, and the directions of the particles. Therefore, the high energy approximation can be achieved very conveniently. Applications of the described method have been performed to the scattering of electrons by electrons or positrons including the exchange of Z0 and Higgs particles.
Quasiclassical trajectory-coherent states of a particle in an arbitrary electromagnetic field
Authors: Bagrov V. G. , Belov V. V. , Ternov I. M.
Published: March 2003

In this paper we show that for a nonrelativistic charged particle moving in an arbitrary external electromagnetic field there exist approximate solutions of the Schrödinger equation, such that the quantum-mechanical averages of the coordinates and the momenta with respect to these states are general exact solutions of the classical Hamiltonian equations. Such states are called trajectory-coherent states. The wave functions of the trajectory-coherent states are obtained by the complex germ method by V. P. Maslov. The simplest properties of these states are studied.
Computation of nonlinear behavior of Hamiltonian systems using Lie algebraic methods
Authors: Dragt Alex J. , Forest Etienne
Published: March 2003

Lie algebraic methods are developed to describe the behavior of trajectories near a given trajectory for general Hamiltonian systems. A procedure is presented for the computation of nonlinear effects of arbitrarily high degree, and explicit formulas are given through effects of degree 5. Expected applications include accelerator design, charged particle beam and light optics, other problems in the general area of nonlinear dynamics, and, perhaps, with suitable modification, the area of S-matrix expansions in quantum field theory.
Generalized canonical transformations for time-dependent systems
Authors: Asorey Manuel , Cariñena José F. , Ibort Luis A.
Published: March 2003

We introduce the concept of generalized canonical transformations as symplectomorphisms of the extended phase space. We prove that any such transformation factorizes in a standard canonical transformation times another one that changes only the time variable. The theory of generating functions as well as that of Hamilton–Jacobi is developed. Some further applications are developed.
Stochastic electrodynamics for the free particle
Authors: de la Peña L. , Jáuregui A.
Published: March 2003

The theory of stochastic electrodynamics is applied to the free particle and to the particle moving in a homogeneous field, leading to a complete temperature- and time-dependent description in phase space. After a transient time, the marginal description in configuration space coincides entirely with quantum mechanics, while the phase-space description is only mathematically related to the Wigner distribution. The Schrödinger equation appears as a natural—though incomplete—means of describing the statistical behavior of the electron under these conditions.
Evolution theorem for a class of perturbed envelope soliton solutions
Authors: Laedke E. W. , Spatschek K. H. , Stenflo L.
Published: March 2003

Envelope soliton solutions of a class of generalized nonlinear Schrödinger equations are investigated. If the quasiparticle number N is conserved, the evolution of solitons in the presence of perturbations can be discussed in terms of the functional behavior of N(η2), where η2 is the nonlinear frequency shift. For ∂η2N >0, the system is stable in the sense of Liapunov, whereas, in the opposite region, instability occurs. The theorem is applied to various types of envelope solitons such as spikons, relatons, and others.
Extension of Fuda’s off-shell analysis to screened Coulomb potentials for arbitrary l and limiting relations
Authors: Dutt Ranabir , Varshni Y. P.
Published: March 2003

The Ecker–Weizel approximation technique is applied to the Schrödinger equation for a class of screened Coulomb potentials (Yukawa, Exponential cosine screened Coulomb and Hulthén) for any arbitrary angular momentum l. We find that the centrifugal term can be combined with the central screening potential to generate an effective Eckart potential with energy dependent strength parameters for which the s-wave Schrödinger equation is exactly solvable. Using this effective s-wave potential in the formalism of Fuda and Whiting for off-shell analysis, we obtain a closed expression for the off-shell Jost solution fS,l (k,q,r) in which k is the on-shell momentum, q is the off-shell momentum and the subscript S means screening. It turns out that for nonzero angular momentum, usual Jost function  fS,l (k,q) can not be defined for finite screening parameter λ. However, we find that the Jost solution, as well as the Jost function defined in the limit λ → 0, show discontinuities at the on-shell point q=k, similar to the observation made by van Haeringen [Phys. Rev. A 18, 56 (1978)] for the s-wave Hulthén potential. For the l=0 case, we obtain explicit expressions for the off-shell and on-shell Jost solutions and Jost functions which possess the limiting behaviors discussed by van Haeringen for the Hulthén potential only. Our results not only extend previous works to higher partial waves, but at the same time indicate that certain limiting properties of the Jost solutions and the Jost functions are generally true for a class of screened Coulomb potentials.
Structural phase transitions in crystals: Broken-symmetry (isotropy) groups
Authors: Jarić Marko V.
Published: March 2003

A systematic procedure ensuring the determination of all isotropy groups of a given representation of a space group is presented for the first time. Isotropy groups play an important role in various areas of theoretical solid state physics. For example, only isotropy groups may occur in a structural phase transition driven by an order parameter belonging to a given representation. The method uses the chain criterion directly on the image of the representation, employing a labeling of the matrices by space group elements. A notion of a substar of a wave vector associated with the representation is central to the method. Finally, as a detailed illustration the method is applied to a structural phase transition in A15 systems driven by an X-point order parameter. The result agrees with previously reported ones.
Convergence of multitime correlation functions in the weak and singular coupling limits
Authors: Dümcke R.
Published: March 2003

For a system coupled to a thermal bath we prove the convergence of the multitime correlation functions of system observables in the weak and singular coupling limits. The limiting correlation functions are given by the quantum regression law. Therefore, our result implies that in the limit the dynamics of the system are governed by a quantum stochastic process in the sense of Lindblad.
A geometrical theory of energy trajectories in quantum mechanics
Authors: Hall Richard L.
Published: March 2003

Suppose f(r) is an attractive central potential of the form f(r)=∑ki=1 g(i)( f(i)(r)), where {f(i)} is a set of basis potentials (powers, log, Hulthén, sech2) and {g(i)} is a set of smooth increasing transformations which, for a given f, are either all convex or all concave. Formulas are derived for bounds on the energy trajectories Enl =Fnl(v) of the Hamiltonian H=−Δ+vf(r), where v is a coupling constant. The transform Λ( f)=F is carried out in two steps: f→f̄→F, where f̄(s) is called the kinetic potential of f and is defined by f̄(s)=inf(ψ,f,ψ) subject to ψ∈𝒟⊆L2(R3), where 𝒟 is the domain of H, ∥ψ∥=1, and (ψ,−Δψ)=s. A table is presented of the basis kinetic potentials { f̄(i)(s)}; the general trajectory bounds F*(v) are then shown to be given by a Legendre transformation of the form (s, f̄*(s)) →(v, F*(v)), where f̄*(s) =∑ki=1g(i)× ( f̄(i)(s)) and F*(v) =mins>0{s+v f̄*(s)}. With the aid of this potential construction set (a kind of Schrödinger Lego), ground-state trajectory bounds are derived for a variety of translation-invariant N-boson and N-fermion problems together with some excited-state trajectory bounds in the special case N=2. This article combines into a single simplified and more general theory the earlier ‘‘potential envelope method’’ and the ‘‘method for linear combinations of elementary potentials.’’
Quantization of spinor fields. III. Fermions on coherent (Bose) domains
Authors: Garbaczewski Piotr
Published: March 2003

A formulation of the c-number classics-quanta correspondence rule for spinor systems requires all elements of the quantum field algebra to be expanded into power series with respect to the generators of the canonical commutation relation (CCR) algebra. On the other hand, the asymptotic completeness demand would result in the (Haag) expansions with respect to the canonical anticommutation relation (CAR) generators. We establish the conditions under which the above correspondence rule can be reconciled with the existence of Haag expansions in terms of asymptotic free Fermi fields. Then, the CAR become represented on the state space of the Bose (CCR) system.
On the dimensional regularization procedure for massless Feynman integrals
Authors: Kalinowski M. W. , Seweryński M. , Szymanowski L.
Published: March 2003

It is shown that the dimensional regularization procedure for massless Feynman integrals, proposed by Capper and Leibbrandt, is unsuitable for practical calculations. This is due to the fact that the procedure yields a logarithmic singularity for some specific massless Feynman integral.
High frequency asymptotic solutions of Yang–Mills and associated fields
Authors: Choquet-Bruhat Yvonne , Greco Antonio
Published: March 2003

We establish the differential equations which rule the propagation of the high-frequency waves, disturbances of a given background, for the coupled Yang–Mills, scalar and spinor field equations. We discuss their interaction.
Expansions around the Gaussian and Lorentzian limits for randomly perturbed spectral lines and their shift, width, and asymmetry
Authors: Royer Antoine
Published: March 2003

A normalized and centered spectral profile P(ω) is conveniently expressed as P(ω)=(2π)−1∫∞−∞ dτexp[−iωτ−∫τ0 dt(τ−t)Ψ(t)], which defines Ψ(t). We consider cases satisfying certain conditions, in particular ‖∫∞0dtΨ(t)‖<∞. A ‘‘broadening strength’’ Λ=ΩΘ is defined, where Ω2 and Θ are the amplitude and characteristic time scale of Ψ(t), respectively. If we let, formally, Λ vary freely, P(ω) tends to a Lorentzian when Λ→0 (weak broadening or strong narrowing limit), and to a Gaussian when Λ→∞ (strong broadening limit). To deal with situations where one of these limit shapes is only approached, for Λ not being small or large enough, we obtain for P(ω) and its shift, width, and asymmetry two kinds of expansions: one in powers of Λ2, starting with the weak broadening limit; the other in powers of Λ−1, starting with the strong broadening limit. Such expansions should allow one to describe spectral profiles over much wider ranges of physical conditions than does the use of just the Lorentzian and Gaussian limit shapes.
A generalization of the construction of Ilamed and Salingaros
Authors: Wene G. P.
Published: March 2003

Ilamed and Salingaros construct the real and complex algebras with three anticommuting elements which can arise in physics. It is shown here that the ‘‘algebra of color’’ can be similarly constructed with six anticommuting elements. As a consequence of this construction, these algebras are all simple, quadratic algebras.
On the Laplace asymptotic expansion of conditional Wiener integrals and the Bender–Wu formula for x2N-anharmonic oscillators
Authors: Davies Ian , Truman Aubrey
Published: March 2003

Rigorous results on the Laplace expansions of conditional Wiener integrals with functional integrands having a finite number of global maxima are established. Applications are given to the Bender–Wu formula for the x2N -anharmonic oscillator.
Generalized Fokker–Planck equation for non-Markovian processes
Authors: Volkov V. S. , Pokrovsky V. N.
Published: March 2003

This paper deals with a system of linear non-Markovian–Langevin equations with memory functions that are not constant in time and a nonzero initial instant of time. A set of statistical means, based on the application of a generalized Furutsu–Novikov formula, was used to derive a generalized Fokker–Planck equation corresponding to this system and holding for both long and short instants of time. Considered as an example is the Brownian motion of a particle in a viscoelastic fluid with a particular relaxation time.
Symmetry and fermion degeneracy on a lattice
Authors: Raszillier H.
Published: March 2003

We consider the general form of the finite difference approximation to the Dirac (Weyl) Hamiltonian on a lattice and investigate systematically the dependence on symmetry of the number of particles described by it. To a lattice with given symmetry, expressed by its crystallographic space group, there corresponds a minimal number of particles, which are associated with prescribed points of momentum space (the unit cell of the reciprocal lattice). We show in tables, using the existing detailed descriptions of the space groups, our results for all the relevant (symmorphic) symmetry groups. Only for lattice Hamiltonians with a momentum dependent mass term can this degeneracy be reduced or eliminated without reducing the symmetry.
Coulomb-modified nuclear scattering. II
Authors: Talukdar B. , Saha S. , Sasakawa T.
Published: March 2003

Exact analytical expressions for the off-shell Jost function fl(k, q) for the Coulomb and Coulomb-distorted nuclear potentials are constructed in terms of the sum of Gaussian hypergeometric functions by using an integral representation for fl(k, q) given by Fuda. It is pointed out that the hypergeometric functions which occur here can be generated by the use of a well-known three-term recurrence relation. With van Haeringen our expressions for fl(k, q) exhibit discontinuity in the on-shell limit q→k. This discontinuous behavior has been attributed to the long range nature of the Coulomb force.
On massive electromagnetism in the null-plane: their limit and ‘‘dual’’ invariance
Authors: Salamó S.
Published: March 2003

The limiting process is developed, stressing the simplicity of the null-plane formulation. A SU(2) ‘‘dual’’ invariance for the dynamical variables is made explicit. The symmetry is not satisfied by the field strengths.
Charged particles in an intense, plane electromagnetic wave: Limitations of the nonrelativistic theory
Authors: Drühl Kai , McIver J. K.
Published: March 2003

We discuss the classical and quantum theory of a charged particle moving in a plane wave electromagnetic field. In the nonrelativistic theory forbidden regions in the momentum spectrum of the particle occur, while no such regions exist in the relativistic theory. The gaps in the momentum spectrum are shown to be an artifact of the nonrelativistic approximation and are caused by the possibility of transforming to a Galilean frame of reference moving at the speed of light in which the particle moves in a time independent periodic potential. A quantitative estimate of the momentum gaps for optical frequencies shows that they are proportional to a very large power n≂105 of the ratio of particle velocity to the speed of light, and hence, beyond the limits of accuracy of the nonrelativistc theory. For this result to hold in the low frequency limit we have to employ a conjecture about the radius of convergence of the perturbation series expansion for eigenvalues of Mathieu’s equation. While no rigorous proof seems to exist at present, this conjecture is well supported both by numerical studies and by our WKB calculations.
Strong fluctuations in the spinor wave optics. Mutual coherence function
Authors: Hillion P. , Quinnez S.
Published: March 2003

In this paper, we apply the spinor wave optics to light beam propagation in a strong turbulent medium. Using the Markov technique, we give the equations for the mutual coherence 4-vector Γμ(z; ρ1, ρ2), μ=0,1,2,3, whose third component Γ3(z; ρ1, ρ2) is the usual mutual coherence function (the three other components have to do with the Poynting vector). Then, scaling transverse and longitudinal distances by the beam waist ω0 and diffraction length l, respectively, an expansion procedure in powers of ω0l−1 is developed. The zeroth-order approximation is discussed, and it is shown that this approximation gives the same mutual coherence function as the paraxial optics. The case of a plane wave is also considered.
The Painlevé property for partial differential equations
Authors: Weiss John , Tabor M. , Carnevale George
Published: March 2003

In this paper we define the Painlevé property for partial differential equations and show how it determines, in a remarkably simple manner, the integrability, the Bäcklund transforms, the linearizing transforms, and the Lax pairs of three well-known partial differential equations (Burgers’ equation, KdV equation, and the modified KdV equation). This indicates that the Painlevé property may provide a unified description of integrable behavior in dynamical systems (ordinary and partial differential equations), while, at the same time, providing an efficient method for determining the integrability of particular systems.
An upper bound on the allowed bands of the Bloch spectrum of one-dimensional Schrödinger operators with periodic potentials
Authors: Meiman N. N.
Published: March 2003

Using the principle of increasing flux and its complement, an upper bound associated with the allowed bands of the Bloch spectrum of one-dimensional Schrödinger operators with periodic potentials is derived. Namely, let a be the period of the potential, E0 the lower bound of the Bloch spectrum and (E2( j−1),E2j−1) the allowed bands ( j=1,2,...). Then π>a[(E2j−1−E0)1/2 −(E2(j−1)−E0)1/2].
Explicit Hilbert space representations of Schrödinger states: Definitions and properties of Stieltjes–Tchebycheff orbitals
Authors: Hermann M. R. , Langhoff P. W.
Published: March 2003

Stieljes–Tchebycheff orbital approximations are described for the discrete and continuun Schrödinger states of self-adjoint (Hamiltonian) operators, and certain of their properties are established and clarified. The nth order orbitals are defined in accordance with corresponding moment-theory approximations to spectral densities as eigenfunctions of the appropriate operator in an n-term Cauchy–Hilbert space. Their eigenvalues are consequently the generalized Gaussian or Radau quadrature points of the associated spectral density function formed by projection with an appropriate test function on the Schrödinger states, and their norms provide the corresponding (reciprocal) quadrature weights. Stable algorithms are described for their construction employing recursive Lanczos and orthogonal-polynomial methods. In finite orders the spatial characteristics of the orbitals correspond to spectral averages in the neighborhoods of the quadrature points over the correct Schrödinger states. The spectral content of an individual orbital is obtained in closed form without reference to the correct underlying Schrödinger states. Convergence (n→∞) is obtained in the discrete spectral region to Schrödinger eigenstates of finite norm, whereas in the essential portion of the spectrum the orbitals converge to scattering states of improper (infinite) norm. Connections with matrix partitioning and optical potential theory are made, indicating that first- order orbitals can provide exact Schrödinger states over local predetermined portions of configuration space. Illustrative computational studies of the regular l-wave spectra of the Coulomb Hamiltonian are provided.
A class of solvable Pauli–Schrödinger Hamiltonians
Authors: Calvo Miguel , Codriansky Simón
Published: March 2003

The Pauli–Schrödinger equation for a spin 1/2, neutral particle with a nonvanishing magnetic moment μ0, interacting with an external scalar potential V and a static magnetic field B, both functions of only one of the coordinates, is solved exactly for four different choices of the potential and the field. By choosing in the examples the coordinate y, we present these solutions in the following cases: (i) V( y)=0, B( y)=(B0sin κy, 0, B0cos κy) where B0 and κ are two arbitrary constants. (ii) V( y)=λ′y2, B( y)=‖αy‖(cos 2κy, 0, sin 2κy) where 6* λ=ℏ2/2m and α,κ arbitrary constants and λ′=λ(μ0/2κλ)2. (iii) V( y)=λ′(c̄1tan αy+c̄2cot αy)2, B( y)=B( y) (sin (2κy+2δ(y)), 0, cos (2κy+2δ( y)) where B2( y)=(c̄1tan αy+c̄2cot αy)2 (+α2/4κ2)(c̄1sec2αy−c̄2csc2αy)2 and tan 2δ( y)=(2κ/λ)((c̄1tan αy+c̄2cot αy) /(c̄1sec2αy−c̄2csc2αy)); c̄1, c̄2, α, and κ are arbitrary constants. (iv) V( y)=λ′(ā tanh z+b̄)2 where z =[(y−cd)/d], B2( y)=(ā tanh z+b̄)2+(ā2/4κ2d2)sech2z, tan 2δ( y)=(2κd/ā)((ā tanh z+b̄)/sech2z); ā, b̄, c, d, and κ are arbitrary constants restricted only by (μ0/2κλ)ā2+ā/d>0. The functions B( y) and δ( y) define the vector B( y) as in (iii). Our method of solution is based on the familiar factorization technique for solving the Schrödinger eigenvalue equation. Several interesting physical results of these solutions are discussed.
Irregular solutions of the s-wave radial Schrödinger equation for van der Waals potentials
Authors: Romo W. J.
Published: March 2003

Simple ideas based on quantum chromodynamics suggest that the strong nuclear force might possess a long-range van der Waals component. Such a component, if it exists, would be most apparent in low-energy interactions between pairs of hadrons, at least one of which is neutral. In this article, as the initial part of a program to set limits on the magnitude of the strength of such a van der Waals interaction, we construct low-energy series expansions of the two irregular solutions f0(±k,r) to the s-wave Schrödinger equation that describes the interaction of a pair of particles interacting through a potential with a van der Waals tail of the form V(r)∼−λℏcrN−10 /rN, for arbitrary values of the potential parameters, λ, r0, and N. Accurate approximation schemes based on the series expansions are developed, and the results are used to obtain an expression for the low-energy elastic cross section. The effects of the van der Waals component of this interaction are clearly evident in the energy dependence of the cross section.
Scattering theory in arrangement channel quantum mechanics
Authors: Evans J. W. , Hoffman D. K. , Kouri D. J.
Published: March 2003

The time-independent scattering theory associated with the non-self-adjoint matrix Hamiltonians H of arrangement channel quantum mechanics is presented in detail first using the 3-particle case as an example. A key feature is the biorthogonality of a suitably constructed set of scattering eigenvectors and duals. Channel space Möller operators, S- and T-matrices are defined and a variety of properties investigated including the way multichannel unitarity is embedded into the theory. Some remarks on the time-dependent theory are also made. A detailed discussion of channel space density matrix scattering theory (of interest, e.g., in reactive kinetic theory) is presented using the Liouville representation. We describe some special cases including the exclusion of breakup and 2×2 choices of three particle H.
Null strings and Bianchi identities
Authors: Torres del Castillo G. F.
Published: March 2003

The results of the theory of complexified V4’s which admit a null string are reexamined by using systematically the Bianchi identities. The formal reasons which permit the integration of the Gμν=0 equations to one differential constraint in the case of the ℋℋ structures are exhibited. Some new results concerning complexified Einstein–Weyl equations in the presence of a null string are given as well.
Orthogonal 2+2 decomposition of space-time
Authors: Smallwood J.
Published: March 2003

A formalism is developed describing the most general 2+2 decomposition of space-time into orthogonal spacelike and timelike 2-surface elements {S} and {T}. No restriction is made on the holonomicity of {S} or {T}, and the formalism is 2-covariant in both {S} and {T}. As an example of the utility of this formalism, the anholonomic 2+2 formulation of the initial value problems in relativity is discussed and the standard holonomic approach is briefly reviewed. Further applications of the formalism are indicated.
A unified theory of the point groups. II. The general projective representations and their application to space groups
Authors: Kim Shoon K.
Published: March 2003

The general expressions of all inequivalent irreducible projective representations of crystallographic and noncrystallographic point groups (except for the icosahedral group) are given in terms of the vector representations of the proper double point groups. Its application to the representations of the space group of wave vector is discussed. Construction of the basis sets is discussed.
Phase conventions consistent with duality between SUn and S
L
Authors: Sullivan John J.
Published: March 2003

The Schur–Weyl construction of irreducible tensors by symmetrized Lth rank exterior products of a defining n-dimensional vector space establishes a duality between the coupling algebra of the unitary group Un and the symmetric group SL . The coupling coefficients shared by these two groups are real and possess symmetries due to complex conjugation in Un and association in SL . The unitary group is factored by its unitary unimodular subgroup SUn≂Un/U1 so it is usual to identify the coupling algebras of Un and SUn by choosing the trivial (phaseless) coupling for U1. This establishes a set of equivalence relations since all pseudoscalar irreducible representations (irreps) of Un subduce onto the scalar irrep of SUn. The question remains can this trivial identification with the coupling algebra of SUn be made consistent with the symmetries which follow from duality? Specifically the symmetries which follow from duality must be consistent with the equivalence relations. We argue such consistency may be obtained. The resulting symmetries place significant restrictions on the resolution of multiplicities and the choice of phase conventions for transformations which in the absence of duality are normally considered independently. We review the usual formulation of a coupling algebra and the identifications which follow from duality using a double coset decomposition of SL . Particular attention is given to the phase transformation introduced by a transposition in order of the two component irreps being coupled. Phase transformations needed for association and complex conjugation are examined to determine the symmetries they must exhibit to be consistent with duality. Two phase transformations needed for complex conjugation, the Derome–Sharp matrix and the 1jm factor, are shown to be related by association. A transposition phase convention and an association phase convention are proposed and shown to be consistent with all the symmetries required by duality.
Models of nonlinear representations and examples of linearization techniques
Authors: Pinczon Georges , Simon Jacques C. H.
Published: March 2003

Examples of classes of nonlinear representations of Lie groups are given. Nonlinear representations which are a perturbation of a unitary representation of the discrete series of SU(1,1) are then proved to be formally linearizable.
Euler angle parametrization of the complex rotation group and its subgroups. I
Authors: Syed Ansaruddin
Published: March 2003

A parametrization of the n-dimensional complex rotation group O+(n, C) by a set of (1/2)n(n−1) complex Euler angles is obtained in exactly the same way as that for the n-dimensional real rotation group O+(n, R). Certain subgroups L̂(n, r) of O+(n, C) are then considered and found isomorphic with the ‘‘generalized Lorentz groups’’ L(n, r), of which the special case L(4, 1) is the usual Lorentz group L of special relativity. Distinguishing features, in the form of reality nature, of Euler angles of L̂(n, r) are obtained for r=1, 2, and it is proved that these lead naturally to the definition of ‘‘real’’ Euler angles of the corresponding L(n, r), which, of course, include the Lorentz group L.
Euler angle parametrization of the complex rotation group and its subgroups. II
Authors: Syed Ansaruddin
Published: March 2003

The work of Paper I is extended here by obtaining the reality nature of Euler angles of the subgroups L̂(n,r) (defined there) of O+(n,C), for arbitrary r, and then showing that these again enable one to define ‘‘real’’ Euler angles of the generalized Lorentz groups L(n,r), r arbitrary.
Geometric construction and properties of some families of solutions of nonlinear partial differential equations. I
Authors: Hermann Robert
Published: March 2003

This series of papers deals with ‘‘The 19th Century Theory of Partial Differential Equations from an Advanced Standpoint.’’ In the treatises of Darboux, Goursat, and Forsythe one finds methods for classifying nonlinear differential equations according to the geometric properties of families of solutions. This work was used by Elie Cartan in his theory of exterior differential systems, and is involved in an indirect way in today’s work on ‘‘nonlinear physics.’’ I plan to present several major themes of the classical work (e.g., ‘‘general,’’ ‘‘singular,’’ and ‘‘complete’’ solutions, ‘‘intermediate integral’’) using geometric methods developed by Cartan, Vessiot, Ehresmann, and Spencer. My aim is to develop this material from a point of view that is both fundamental and directed toward its ultimate application. Another possible utilization is in the development of the symbolic computation computer systems such as MACSYMA and REDUCE. This first paper concentrates on a description of what is meant by a ‘‘general solution,’’ and presents the classical Lagrange–Charpit method for constructing ‘‘general’’ solutions by means of a ‘‘complete solution.’’ The Monge–Ampère equation is also treated in modernized form. A new geometric concept, that of a Lagrange–Vessiot submersion, is isolated from the classical literature.
Complementary group with respect to SO(n)
Authors: Couvreur G. , Deenen J. , Quesne C.
Published: March 2003

We look for a complementary group with respect to SO(n) within either irreducible representation 〈(1/2)dn〉 or 〈(1/2)dn−1(3/2)〉 of the group Sp(2dn,R) of linear canonical transformations in a 2dn-dimensional phase space. We prove that: (i) such a group is Sp(2d,R) when n=2q+1 or n=2q>2d; (ii) it is SU(d,d) when n=2 and d>1; (iii) it does not exist when n=2, d=1, or 2<n=2q≤2d.
Some special SU(3)⊇R(3) Wigner coefficients and their application
Authors: Hecht K. T. , Suzuki Y.
Published: March 2003

Bargmann space expansions of oscillator functions are used to derive analytic expressions for SU(3)⊇R(3) Wigner coefficients for the couplings (λ10)×(0μ2)→(λ3 μ3)L3=0 and (λ10)×(λ20)→(λ3 μ3)L3=0, with arbitrary (λ3 μ3). These lead to expansions useful in nuclear cluster problems and are used to give a simple form for the SU(3)-irreducible tensor expansion of a scalar two-body interaction, an application which motivated this investigation.
Reciprocal transformation for one-dimensional conservation equations
Authors: Rosen Gerald
Published: March 2003

One-dimensional conservation equations (OCE) of the form ∂n/∂t+∂f/∂x=0 with n=n(x,t)>0 and f=f(n,∂n/∂x, ∂2n/∂x2,⋅⋅⋅) admit a symmetric reciprocal transformation x→x*(x,t), n→n*(x*,t)≡n−1, f→f*≡−n−1f, which produces an equivalent OCE for n* in x* space. Certain OCE of contemporary interest are reciprocal invariant in the sense that f*=f(n*, ∂n*/∂x*, ∂2n*/∂x*2,⋅⋅⋅). There also exists a class of essentially nonlinear OCE for which the reciprocal transformation produces a linear OCE, and thus equations in this class are solvable analytically.
Lie transformations, nonlinear evolution equations, and Painlevé forms
Authors: Lakshmanan M. , Kaliappan P.
Published: March 2003

We present the results of a systematic investigation of invariance properties of a large class of nonlinear evolution equations under a one-parameter continuous (Lie) group of transformations. It is shown that, in general, the corresponding invariant variables (the subclass of which is the usual similarity variables) lead to ordinary differential equations of Painlevé type in the case of inverse scattering transform solvable equations, as conjectured by Ablowitz, Ramani, and Segur. This is found to be also true for certain higher spatial dimensional versions such as the Kadomtsev–Petviashivilli, two dimensional sine–Gordon, and Ernst equations. For the nonsolvable equations considered here this invariance study leads to ordinary differential equations with movable critical points.
Weak dispersion-free states and the hidden variables hypothesis
Authors: Pták Pavel
Published: March 2003

We investigate dispersion-free states which are additive only on the pairs containing a central element (central-absolutely compatible). We show that any logic possesses plenty of such states, in fact, as many as a certain Boolean algebra. The latter result matches the hidden variables conjecture.
Path integration of an action related to an electron gas in a random potential
Authors: Dhara A. K. , Khandekar D. C. , Lawande S. V.
Published: March 2003

Path integration of an action related to an electron gas in a random potential is performed within the framework of Feynman’s polygonal path approach. The exact propagator obtained is simply related to the harmonic oscillator propagator. The integration is direct and does not require the knowledge of an auxiliary measure or the artificial coupling of the system to the external forces.
A partial inner product space of analytic functions for resonances
Authors: Horwitz L. P. , Katznelson E.
Published: March 2003

Generalized Hilbert spaces D(α,β) are defined using analytic continuation of Hardy class functions into a wedge bounded by the angles α,β. Eigenfunctions of isolated complex eigenvalues may be found in D(α,β) for operators that have a self-adjoint representation in L2. These eigenvalues correspond to resonances in the associated decay problem. A bilinear form between D(α,β) and D(−β,−α) is defined, which has some of the properties of a Hilbert space scalar product, and it is shown that this form can be used to define a variational principle to obtain the eigenvalue equations.