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Dec 1999

Volume 40, Issue 12, pp. 6133-6707

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Quantum description of rigidly or adiabatically constrained four-particle systems and supersymmetry

E. Baloïtcha and M. N. Hounkonnou

J. Math. Phys. 40, 6133 (1999); http://dx.doi.org/10.1063/1.533082 (12 pages) | Cited 7 times

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A general formalism for the quantum description of many-body systems is developed, using the principal-axis hyperspherical parametrization of coordinates. This formalism is applied to four-particle systems for which the exact kinetic energy operator is derived using a model constraints and dynamical constraints. Then, using the supersymmetry and shape invariance approach, we obtained in a closed form the eigenvalues and eigenfunctions of a wide class of noncentral potentials for the adiabatically constrained four-particle systems. © 1999 American Institute of Physics.
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45.50.Jf Few- and many-body systems
03.65.Db Functional analytical methods
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra

Coulomb wave functions with complex values of the variable and the parameters

Aleksander Dzieciol, Staffan Yngve, and Per Olof Fröman

J. Math. Phys. 40, 6145 (1999); http://dx.doi.org/10.1063/1.533083 (22 pages) | Cited 7 times

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The motivation for the present paper lies in the fact that the literature concerning the Coulomb wave functions FL(η,ρ) and GL(η,ρ) is a jungle in which it may be hard to find a safe way when one needs general formulas for the Coulomb wave functions with complex values of the variable ρ and the parameters L and η. For the Coulomb wave functions and certain linear combinations of these functions we discuss the connection with the Whittaker function, the Coulomb phase shift, Wronskians, reflection formulas (L→−L−1), integral representations, series expansions, circuital relations (ρ→ρe±iπ) and asymptotic formulas on a Riemann surface for the variable ρ. The parameters L and η are allowed to assume complex values. © 1999 American Institute of Physics.
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02.30.-f Function theory, analysis
03.65.Ge Solutions of wave equations: bound states
03.65.Db Functional analytical methods

Phase-integral formulas for Coulomb wave functions with complex values of the variable and the parameters

Aleksander Dzieciol, Staffan Yngve, and Per Olof Fröman

J. Math. Phys. 40, 6167 (1999); http://dx.doi.org/10.1063/1.533084 (11 pages) | Cited 1 time

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Phase-integral formulas for the Coulomb wave functions FL(η,ρ) and GL(η,ρ) and certain linear combinations of these functions, with complex values of the variable ρ and the parameters L and η, are obtained explicitly up to the fifth order of the phase-integral approximation for two different choices of the base function. © 1999 American Institute of Physics.
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02.30.Jr Partial differential equations
02.30.Cj Measure and integration

Collective dynamics of solitons and inequivalent quantizations

J. P. Garrahan and M. Kruczenski

J. Math. Phys. 40, 6178 (1999); http://dx.doi.org/10.1063/1.533085 (11 pages) | Cited 2 times

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The collective dynamics of solitons with a coset space G/H as moduli space is studied. It is shown that the collective band for a vibrational state is given by the inequivalent coset space quantization corresponding to the representation of H carried by the vibration. To leading order the collective dynamics is free motion in G/H coupled to background gauge fields determined by the vibrational state. © 1999 American Institute of Physics.
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11.10.Lm Nonlinear or nonlocal theories and models
11.15.-q Gauge field theories

The osp(1,2)-covariant Lagrangian quantization of reducible massive gauge theories

B. Geyer, P. M. Lavrov, and D. Mülsch

J. Math. Phys. 40, 6189 (1999); http://dx.doi.org/10.1063/1.533086 (20 pages) | Cited 1 time

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The osp(1,2)-covariant Lagrangian quantization of irreducible gauge theories is generalized to L-stage reducible theories. The dependence of the generating functional of Green’s functions on the choice of gauge in the massive case is discussed and Ward identities related to osp(1,2) symmetry are given. Massive first-stage theories with closed gauge algebra are studied in detail. The generalization of the Chapline–Manton model and topological Yang–Mills theory to the case of massive fields are considered as examples. © 1999 American Institute of Physics.
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11.15.-q Gauge field theories
11.10.Ef Lagrangian and Hamiltonian approach
02.40.Pc General topology
02.10.-v Logic, set theory, and algebra
02.30.-f Function theory, analysis

Ground states of a model in nonrelativistic quantum electrodynamics. I

Fumio Hiroshima

J. Math. Phys. 40, 6209 (1999); http://dx.doi.org/10.1063/1.533087 (14 pages) | Cited 6 times

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The system of a one charged nonrelativistic particle with external potentials minimally coupled to a massless quantized radiation field is considered. An ultraviolet cutoff is imposed on the quantized radiation field and the charged particle has spin 1/2. The class of external potentials considered in this paper contains Coulomb potentials. It is shown that the ground states of the system exist provided that a coupling constant is in a region. © 1999 American Institute of Physics.
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12.20.Ds Specific calculations

Spherically symmetric solutions of the SU(N) Skyrme models

T. Ioannidou, B. Piette, and W. J. Zakrzewski

J. Math. Phys. 40, 6223 (1999); http://dx.doi.org/10.1063/1.533088 (11 pages) | Cited 15 times

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Recently we have presented an ansatz which allows us to construct skyrmion fields from the harmonic maps of S2 to CPN−1. In this paper we examine this construction in detail and use it to construct, in an explicit form, new static spherically symmetric solutions of the SU(N) Skyrme models. We also discuss some properties of these solutions. © 1999 American Institute of Physics.
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11.10.Lm Nonlinear or nonlocal theories and models
11.30.Ly Other internal and higher symmetries

Two algebraic properties of thermal quantum field theories

Christian D. Jäkel

J. Math. Phys. 40, 6234 (1999); http://dx.doi.org/10.1063/1.533089 (11 pages)

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We establish the Schlieder and the Borchers property for thermal field theories. In addition, we provide some information on the commutation and localization properties of projection operators. © 1999 American Institute of Physics.
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11.10.Wx Finite-temperature field theory
02.10.-v Logic, set theory, and algebra

Nonrelativistic quantum Hamiltonian for Lorentz violation

V. Alan Kostelecký and Charles D. Lane

J. Math. Phys. 40, 6245 (1999); http://dx.doi.org/10.1063/1.533090 (9 pages) | Cited 79 times

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A method is presented for deriving the nonrelativistic quantum Hamiltonian of a free massive fermion from the relativistic Lagrangian of the Lorentz-violating standard-model extension. It permits the extraction of terms at arbitrary order in a Foldy–Wouthuysen expansion in inverse powers of the mass. The quantum particle Hamiltonian is obtained and its nonrelativistic limit is given explicitly to third order. © 1999 American Institute of Physics.
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12.60.-i Models beyond the standard model
03.65.Pm Relativistic wave equations
11.10.Ef Lagrangian and Hamiltonian approach
11.30.Cp Lorentz and Poincaré invariance

Casimir energy of a ball and cylinder in the zeta function technique

G. Lambiase, V. V. Nesterenko, and M. Bordag

J. Math. Phys. 40, 6254 (1999); http://dx.doi.org/10.1063/1.533091 (12 pages) | Cited 28 times

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A simple method is proposed to construct the spectral zeta functions required for calculating the electromagnetic vacuum energy with boundary conditions given on a sphere or on an infinite cylinder. When calculating the Casimir energy in this approach no exact divergencies appear and no renormalization is needed. The starting point of the consideration is the representation of the zeta functions in terms of contour integral, further the uniform asymptotic expansion of the Bessel function is essentially used. After the analytic continuation, needed for calculating the Casimir energy, the zeta functions are presented as infinite series containing the Riemann zeta function with rapidly falling down terms. The spectral zeta functions are constructed exactly for a material ball and infinite cylinder placed in a uniform endless medium under the condition that the velocity of light does not change when crossing the interface. As a special case, perfectly conducting spherical and cylindrical shells are also considered in the same line. In this approach one succeeds, specifically, in justifying, in mathematically rigorous way, the appearance of the contribution to the Casimir energy for cylinder which is proportional to ln(2π). © 1999 American Institute of Physics.
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11.10.-z Field theory
03.70.+k Theory of quantized fields
02.30.Gp Special functions

Quasilinearization method and its verification on exactly solvable models in quantum mechanics

V. B. Mandelzweig

J. Math. Phys. 40, 6266 (1999); http://dx.doi.org/10.1063/1.533092 (26 pages) | Cited 31 times

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The proof of the convergence of the quasilinearization method of Bellman and Kalaba, whose origin lies in the theory of linear programming, is extended to large and infinite domains and to singular functionals in order to enable the application of the method to physical problems. This powerful method approximates solution of nonlinear differential equations by treating the nonlinear terms as a perturbation about the linear ones, and is not based, unlike perturbation theories, on existence of some kind of small parameter. The general properties of the method, particularly its uniform and quadratic convergence, which often also is monotonic, are analyzed and verified on exactly solvable models in quantum mechanics. Namely, application of the method to scattering length calculations in the variable phase method shows that each approximation of the method sums many orders of the perturbation theory and that the method reproduces properly the singular structure of the exact solutions. The method provides final and reasonable answers for infinite values of the coupling constant and is able to handle even super singular potentials for which each term of the perturbation theory is infinite and the perturbation expansion does not exist. © 1999 American Institute of Physics.
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03.65.Db Functional analytical methods
02.30.-f Function theory, analysis
03.65.Nk Scattering theory

Picard–Fuchs equations and Whitham hierarchy in N = 2 supersymmetric SU(r+1) Yang–Mills theory

Yűji Ohta

J. Math. Phys. 40, 6292 (1999); http://dx.doi.org/10.1063/1.533093 (10 pages) | Cited 2 times

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In general, Whitham dynamics involves infinitely many parameters called Whitham times, but in the context of N = 2 supersymmetric Yang–Mills theory it can be regarded as a finite system by restricting the number of Whitham times appropriately. For example, in the case of SU(r+1) gauge theory without hypermultiplets, there are r Whitham times and they play an essential role in the theory. In this situation, the generating meromorphic one-form of the Whitham hierarchy on the Seiberg–Witten curve is represented by a finite linear combination of meromorphic one-forms associated with these Whitham times, but it turns out that there are various differential relations among these differentials. Since these relations can be written only in terms of the Seiberg–Witten one-form, their consistency conditions are found to give the Picard–Fuchs equations for the Seiberg–Witten periods. © 1999 American Institute of Physics.
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11.15.-q Gauge field theories
11.30.Pb Supersymmetry
11.30.Hv Flavor symmetries
11.30.Ly Other internal and higher symmetries
02.30.-f Function theory, analysis

Wigner–Weyl correspondence and semiclassical quantization in spherical coordinates

Bill Poirier

J. Math. Phys. 40, 6302 (1999); http://dx.doi.org/10.1063/1.533094 (17 pages) | Cited 1 time

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The Wigner–Weyl quantum-to-classical correspondence rule is nonunique with respect to coordinate choice. This ambiguity can be exploited to improve the accuracy of semiclassical approximations. For instance, the well-known Langer modification was recently derived by applying a coordinate transformation to the radial Schrödinger equation prior to using the Wigner–Weyl rule—albeit only by presuming exact quantum solutions for all nonradial degrees of freedom [J. J. Morehead, J. Math. Phys. 36, 5431 (1995)]. In this paper, the full classical Hamiltonian is derived in all degrees of freedom, using a (hyper)spherical coordinate Wigner–Weyl correspondence with a Langer-like modification of polar angles. For central force Hamiltonians, the new result is radially equivalent to that of Langer, and to the standard Cartesian form. The new correspondence is superior with respect to all angular momentum operators however, in that the resultant semiclassical eigenvalues are exact—a desirable goal, evidently achieved here for the first time. © 1999 American Institute of Physics.
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03.65.Sq Semiclassical theories and applications
03.65.Ge Solutions of wave equations: bound states
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra

On separable Schrödinger equations

Renat Zhdanov and Alexander Zhalij

J. Math. Phys. 40, 6319 (1999); http://dx.doi.org/10.1063/1.533095 (20 pages) | Cited 8 times

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We classify (1+3)-dimensional Schrödinger equations for a particle interacting with the electromagnetic field that are solvable by the method of separation of variables. As a result, we get 11 classes of the vector potentials of the electromagnetic field A(t,math) = (A0(t,math),math(t,math)) providing separability of the corresponding Schrödinger equations. It is established, in particular, that the necessary condition for the Schrödinger equation to be separable is that the magnetic field must be independent of the spatial variables. Next, we prove that any Schrödinger equation admitting variable separation into second-order ordinary differential equations can be reduced to one of the 11 separable Schrödinger equations mentioned above and carry out variable separation in the latter. Furthermore, we apply the results obtained for separating variables in the Hamilton–Jacobi equation. © 1999 American Institute of Physics.
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03.65.Ge Solutions of wave equations: bound states
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On the explicit solutions of the elliptic Calogero system

L. Gavrilov and A. M. Perelomov

J. Math. Phys. 40, 6339 (1999); http://dx.doi.org/10.1063/1.533096 (14 pages) | Cited 5 times

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Let q1,q2,…,qN be the coordinates of N particles on the circle, interacting with the integrable potential ∑j<kN℘(qj−qk), where ℘ is the Weierstrass elliptic function. We show that every symmetric elliptic function in q1,q2,…,qN is a meromorphic function in time. We give explicit formulas for these functions in terms of genus N−1 theta functions. © 1999 American Institute of Physics.
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45.50.Jf Few- and many-body systems
02.30.Sa Functional analysis

SU(N) skyrmions and harmonic maps

T. Ioannidou, B. Piette, and W. J. Zakrzewski

J. Math. Phys. 40, 6353 (1999); http://dx.doi.org/10.1063/1.533097 (13 pages) | Cited 13 times

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Harmonic maps from S2 to CPN−1 are introduced to construct low-energy configurations of the SU(N) Skyrme model. We show that one of such maps gives an exact, topologically trivial, solution of the SU(3) model. We study various properties of these maps and show that, in general, their energies are only a little higher than the energies of the corresponding SU(2) embeddings. Moreover, we show that the baryon and energy densities of the SU(3) configurations with baryon number B = 3−6 are more symmetrical than their SU(2) analogs, thus suggesting that there exist solutions of the model with these symmetries. We also show that any SU(2) solution embedded into the SU(4) Skyrme model becomes a topologically trivial solution of this model. © 1999 American Institute of Physics.
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11.30.Ly Other internal and higher symmetries
11.10.Lm Nonlinear or nonlocal theories and models

Quasi-Lagrangian systems of Newton equations

Stefan Rauch-Wojciechowski, Krzysztof Marciniak, and Hans Lundmark

J. Math. Phys. 40, 6366 (1999); http://dx.doi.org/10.1063/1.533098 (33 pages) | Cited 16 times

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Systems of Newton equations of the form math = −½A−1(q)∇k with an integral of motion quadratic in velocities are studied. These equations generalize the potential case (when A = I, the identity matrix) and they admit a curious quasi-Lagrangian formulation which differs from the standard Lagrange equations by the plus sign between terms. A theory of such quasi-Lagrangian Newton (qLN) systems having two functionally independent integrals of motion is developed with focus on two-dimensional systems. Such systems admit a bi-Hamiltonian formulation and are proved to be completely integrable by embedding into five-dimensional integrable systems. They are characterized by a linear, second-order partial differential equation PDE which we call the fundamental equation. Fundamental equations are classified through linear pencils of matrices associated with qLN systems. The theory is illustrated by two classes of systems: separable potential systems and driven systems. New separation variables for driven systems are found. These variables are based on sets of nonconfocal conics. An effective criterion for existence of a qLN formulation of a given system is formulated and applied to dynamical systems of the Hénon–Heiles type. © 1999 American Institute of Physics.
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45.20.Jj Lagrangian and Hamiltonian mechanics
45.05.+x General theory of classical mechanics of discrete systems
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra
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Relativistic gas: Moment equations and maximum wave velocity

Guy Boillat and Tommaso Ruggeri

J. Math. Phys. 40, 6399 (1999); http://dx.doi.org/10.1063/1.533099 (8 pages) | Cited 3 times

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For a rarefied relativistic gas we consider the N-moment equations associated with the relativistic Boltzmann–Chernikov equation and we require the compatibility with the entropy principle thus obtaining a closed symmetric hyperbolic system. This interesting form permits one to deduce a lower and an upper bound for the maximum velocity of a wave propagating in a monoatomic or a degenerate gas of fermions or bosons and to prove that when this number N increases this velocity tends to the speed of light. © 1999 American Institute of Physics.
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47.75.+f Relativistic fluid dynamics
47.45.-n Rarefied gas dynamics
51.30.+i Thermodynamic properties, equations of state
05.30.Fk Fermion systems and electron gas
05.30.Jp Boson systems

Integral representations of thermodynamic 1PI Green’s functions in the world-line formalism

Haru-Tada Sato

J. Math. Phys. 40, 6407 (1999); http://dx.doi.org/10.1063/1.533100 (23 pages) | Cited 2 times

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The issue discussed is a thermodynamic version of the Bern–Kosower master amplitude formula, which contains all necessary one-loop Feynman diagrams. It is demonstrated how the master amplitude at finite values of temperature and chemical potential can be formulated within the framework of the world-line formalism. In particular we present an elegant method of how to introduce a chemical potential for a loop in the master formula. Various useful integral formulas for the master amplitude are then obtained. The nonanalytic property of the master formula is also derived in the zero temperature limit with the value of chemical potential kept finite. © 1999 American Institute of Physics.
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05.70.Ce Thermodynamic functions and equations of state
02.30.Rz Integral equations
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A field theory approach to Lindstedt series for hyperbolic tori in three time scales problems

G. Gallavotti, G. Gentile, and V. Mastropietro

J. Math. Phys. 40, 6430 (1999); http://dx.doi.org/10.1063/1.533101 (43 pages) | Cited 1 time

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Interacting systems consisting of two rotators and a pendulum are considered, in a case in which the uncoupled systems have three very different characteristic time scales. The abundance of unstable quasiperiodic motions in phase space is studied via the Lindstedt series regarded as a sum of Feynman graphs and studied with renormalization group techniques based on Eliasson’s work on KAM tori. The result is a strong improvement, compared to our previous results, on the domain of validity of bounds that imply existence of invariant tori, large homoclinic angles, long heteroclinic chains, and drift-diffusion in phase space. © 1999 American Institute of Physics.
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45.05.+x General theory of classical mechanics of discrete systems
02.30.Lt Sequences, series, and summability

On construction of recursion operators from Lax representation

Metin Gürses, Atalay Karasu, and Vladimir V. Sokolov

J. Math. Phys. 40, 6473 (1999); http://dx.doi.org/10.1063/1.533102 (18 pages) | Cited 27 times

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In this work we develop a general procedure for constructing the recursion operators for nonlinear integrable equations admitting Lax representation. Several new examples are given. In particular, we find the recursion operators for some KdV-type systems of integrable equations. © 1999 American Institute of Physics.
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02.30.Jr Partial differential equations
02.30.Tb Operator theory

Formal variable separation approach for nonintegrable models

Sen-yue Lou and Li-Li Chen

J. Math. Phys. 40, 6491 (1999); http://dx.doi.org/10.1063/1.533103 (10 pages) | Cited 49 times

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Using a formal variable separation approach, a nonlinear partial differential equation can be solved by solving ordinary different equations or even algebraic equations. Taking the KdV–Burgers and modified KdV–Burgers equations with background interaction as simple examples, some explicit solitary wave solutions which are induced by background source and nonlinearity or dispersion are obtained. © 1999 American Institute of Physics.
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02.30.Jr Partial differential equations

KP constraints from reduced multi-component hierarchies

R. Willox and I. Loris

J. Math. Phys. 40, 6501 (1999); http://dx.doi.org/10.1063/1.533104 (25 pages) | Cited 2 times

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The (m-vector) k-constrained Kadomtsev–Petviashvili (KP) hierarchy is shown to be a “pseudo”-reduction of the (m+1)-component KP hierarchy. To facilitate the implementation of this reduction on the level of the solutions, the typical multi-component KP solutions are mapped onto solutions of a Toda molecule-type equation from which (Wronskian and Grammian) solutions for the constrained KP hierarchy follow. The reduction of the associated linear systems is discussed and its importance for the choice of bilinear representation of the reduced systems is explained. © 1999 American Institute of Physics.
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02.30.Jr Partial differential equations
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)

Separation of variables for soliton equations via their binary constrained flows

Yunbo Zeng and Wen-Xiu Ma

J. Math. Phys. 40, 6526 (1999); http://dx.doi.org/10.1063/1.533105 (32 pages) | Cited 14 times

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Binary constrained flows of soliton equations admitting 2×2 Lax matrices have 2N degrees of freedom, which is twice as many degrees of freedom than in the case of monoconstrained flows. By using the normal method, their Lax matrices directly give rise to first N pairs of canonical separated variables for their separation of variables. We propose a new method to introduce the other N pairs of canonical separated variables and additional separated equations. The Jacobi inversion problems for binary constrained flows are established. Finally, the factorization of soliton equations by two commuting binary constrained flows and the separability of binary constrained flows enable us to construct the Jacobi inversion problems for some soliton hierarchies. © 1999 American Institute of Physics.
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05.45.Yv Solitons
02.10.Ud Linear algebra
02.10.Xm Multilinear algebra
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Cosmological and spherically symmetric solutions with intersecting p-branes

V. D. Ivashchuk and V. N. Melnikov

J. Math. Phys. 40, 6558 (1999); http://dx.doi.org/10.1063/1.533106 (19 pages) | Cited 9 times

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Multidimensional model describing the cosmological evolution and/or spherically symmetric configuration with n+1 Einstein spaces in the theory with several scalar fields and forms is considered. When electro-magnetic composite p-brane ansatz is adopted, n “internal” spaces are Ricci-flat, one space M0 has a nonzero curvature, and all p-branes do not “live” in M0, a class of exact solutions is obtained if certain block-orthogonality relations on p-brane vectors are imposed. A subclass of spherically symmetric solutions (containing nonextremal p-brane black holes) is considered. Post-Newtonian parameters are calculated. © 1999 American Institute of Physics.
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11.25.-w Strings and branes
98.80.Cq Particle-theory and field-theory models of the early Universe (including cosmic pancakes, cosmic strings, chaotic phenomena, inflationary universe, etc.)
04.70.-s Physics of black holes
04.20.Jb Exact solutions
04.40.Nr Einstein-Maxwell spacetimes, spacetimes with fluids, radiation or classical fields
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