Table of contents

We would like to thank all referees for the work they have done for us in 2003. Effective peer review is essential to the success of the journal and so we greatly appreciate the time and effort that referees give up on behalf of the journal. We hope that we can continue this valued collaboration into the future.

With the help of our referees the journal has grown substantially in recent years. Despite a high rejection rate, our published content has grown by more than 40% in the last five years. Practical considerations mean that we cannot continue to grow the journal at this rate. Accordingly, we will now have to raise the quality threshold of the journal by rejecting those marginal quality papers which previously may have been accepted.

With immediate effect we are asking all our referees to be particularly strict when judging a paper. In particular, please could you implement the following criteria rigorously:

"To be acceptable for publication in J. Phys. A papers must make significant, original and correct contributions to one or more topics within the scope of the journal. Mathematical papers should be clearly motivated by actual or potential application to physical phenomena. The length of a paper should be justified by its contents."

For full details of our scope (see also the subject classification) and refereeing requirements please see the journal homepage (www.iop.org/journals/jpa)

There is a whole range of emergent phenomena in a complex network such as robustness, adaptiveness, multiple-equilibrium, hysteresis, oscillation and feedback. Those non-equilibrium behaviours can often be described by a set of stochastic differential equations. One persistent important question is the existence of a potential function. Here we demonstrate that a dynamical structure built into stochastic differential equation allows us to construct such a global optimization potential function. We present an explicit construction procedure to obtain the potential and relevant quantities. In the procedure no reference to the Fokker–Planck equation is needed. The availability of the potential suggests that powerful statistical mechanics tools can be used in nonequilibrium situations.

We consider the semiclassical limit of the spectral form factor K(τ) of fully chaotic dynamics. Starting from the Gutzwiller-type double sum over classical periodic orbits we set out to recover the universal behaviour predicted by random-matrix theory, both for dynamics with and without time reversal invariance. For times smaller than half the Heisenberg time T_H ∝ ℏ^−f+1, we extend the previously known τ-expansion to include the cubic term. Beyond confirming the random-matrix behaviour of individual spectra, the virtue of that extension is that the 'diagrammatic rules' come in sight which determine the families of orbit pairs responsible for all orders of the τ-expansion.

We investigate the non-commutative (NC) field theory approach to the vortex liquid system restricted to the lowest Landau level (LLL) approximation. NC field theory effectively takes care of the phase space reduction of the LLL physics in a ⋆-product form and introduces a new gauge invariant form of a quartic potential of the order parameter in the Ginzburg–Landau (GL) free energy. This new quartic interaction coupling term has a non-trivial equivalence relation with that obtained by Brézin, Nelson and Thiaville in the usual GL framework. The consequence of the equivalence is discussed.

We give improved bounds for the connective constant of the hexagonal lattice. The lower bound is found by using Kesten's method of irreducible bridges and by determining generating functions for bridges on one-dimensional lattices. The upper bound is obtained as the largest eigenvalue of a certain transfer matrix. Using a relation between the hexagonal and the (3.12²) lattices, we also give bounds for the connective constant of the latter lattice.

We propose a stochastic model for the intersection of two urban streets. The vehicular traffic at the intersection is controlled by a set of traffic lights which can be operated subject to fix-time as well as traffic adaptive schemes. Vehicular dynamics is simulated within the framework of the probabilistic cellular automata and the delay experienced by the traffic at each individual street is evaluated for specified time intervals. Minimizing the total delay of both streets gives rise to the optimum signalization of traffic lights. We propose some traffic responsive signalization algorithms which are based on the concept of cut-off queue length and cut-off density.

A binary lattice gas model that allows for multiple occupancy of lattice sites, inspired by recent coarse-grained descriptions of solutions of interacting polymers, is investigated by combining the steepest descent approximation with an exploration of the multidimensional energy landscape and by Gibbs ensemble Monte Carlo simulations. The one-component version of the model, involving on site and nearest-neighbour interactions, is shown to exhibit microphase separation into two sub-lattices with different mean occupation numbers. The symmetric two-component version of the multiple occupancy lattice gas is shown to exhibit a demixing transition into two phases above a critical mean occupation number.

The two-time non-equilibrium correlation and response functions in 1D kinetic classical spin systems with non-conserved dynamics and quenched to their zero-temperature critical point are studied. The exact solution of the kinetic Ising model with Glauber dynamics for a wide class of initial states allows for an explicit test of the universality of the non-equilibrium limit fluctuation–dissipation ratio X_∞. It is shown that the value of X_∞ depends on whether the initial state has finitely many domain walls or not and thus two distinct dynamic universality classes can be identified in this model. Generic 1D kinetic spin systems with non-conserved dynamics fall into the same universality classes as the kinetic Glauber–Ising model provided the dynamics is invariant under the C-symmetry of simultaneous spin and magnetic-field reversal. While C-symmetry is satisfied for magnetic systems, it need not be for lattice gases which may therefore display hitherto unexplored types of non-universal kinetics.

The phase diagram of the monomer–trimer model has already been studied on the basis of (a thermal) Langmuir–Hinshelwood (LH) mechanism, the details of which can be found in the literature. Here, we have studied the effect of the Eley–Rideal (ER) mechanism on the phase diagram of this system. With the introduction of the ER process, the continuous transition of the LH model has been eliminated, while reactive window width and the production rate have been increased. The production rate can be represented by simple mathematical equations. The effect of monomer diffusion and desorption has also been studied for both models. Results are compared and some interesting observations are reported.

For a noninteracting Bose gas with a fixed one-body Hamiltonian H⁰ independent of the number of particles, we derive the inequalities ⟨N_i⟩_N < ⟨N_i⟩_N+1, ⟨N_iN_j⟩_N < ⟨N_i⟩_N⟨N_j⟩_N for i ≠ j, ∂⟨N₀⟩_N/∂β > 0 and ⟨N_i⟩⁺_N < ⟨N_i⟩_N. Here N_i is the occupation number of the ith eigenstate of H⁰, β is the inverse temperature and the superscript + refers to adding an extra level to those of H⁰. The results follow from the convexity of the N-particle free energy as a function of N.

Using two-electron spin transpositions within BCS pairs, it is shown that an efficient block-diagonal decomposition of the Hamiltonian matrices associated with general pairing models is possible. The ratio of the dimension of the full Hilbert space to the size of the largest submatrix obtained grows exponentially with the number of electron pairs. This remarkable result holds for any electron filling. The approach is exemplified on the usual BCS model.

In this paper we provide an approach for identifying certain mixture representations of some invariant measures of interacting stochastic systems. This is related to the problem of ergodicity of certain extremal invariant measures that are translation invariant. Corresponding to these, results concerning the existence of invariant measures and certain weak convergence of the systems are also provided.

The stochastic dynamics of a classical parametric oscillator driven by coloured noise with exponential correlation is studied both by analytical and numerical methods. The critical strength of the parametric modulation at which the motion of the oscillator becomes unbounded, and the dependence of this critical strength on the magnitude and the correlation length of the noise, are investigated. The results show that the critical strength does not depend on either the strength or the correlation length of the coloured noise, but that the squeezing induced by the parametric oscillation is enhanced by increasing the correlation length of the noise.

Formulae expressing explicitly the Jacobi coefficients of a general-order derivative (integral) of an infinitely differentiable function in terms of its original expansion coefficients, and formulae for the derivatives (integrals) of Jacobi polynomials in terms of Jacobi polynomials themselves are stated. A formula for the Jacobi coefficients of the moments of one single Jacobi polynomial of certain degree is proved. Another formula for the Jacobi coefficients of the moments of a general-order derivative of an infinitely differentiable function in terms of its original expanded coefficients is also given. A simple approach in order to construct and solve recursively for the connection coefficients between Jacobi–Jacobi polynomials is described. Explicit formulae for these coefficients between ultraspherical and Jacobi polynomials are deduced, of which the Chebyshev polynomials of the first and second kinds and Legendre polynomials are important special cases. Two analytical formulae for the connection coefficients between Laguerre–Jacobi and Hermite–Jacobi are developed.

We propose a Mathematica package which allows us to determine all the classes of three-dimensional subalgebras (TDS) of a simple Lie algebra. The programming of the package is based on the theory of semisimple subalgebras of simple Lie algebras. We summarize the main points of this theory, which leads to an algorithm for the construction of the classes of TDS. A particular emphasis is laid on the construction of the exceptional TDS of D_n. The package Decompositions.m implements this algorithm to give all the TDS of the classical simple Lie algebras, principal and nonprincipal. The package provides several functions which characterize the three-dimensional embeddings such as, for example, the set of spins of a decomposition, the defining vector, the pi-system of roots, the generators of the TDS and the generators of each decomposition.

We investigate the two-dimensional Aharonov–Bohm operator , where Γ is a smooth loop and A is the vector potential which corresponds to the Aharonov–Bohm potential. The asymptotics of negative eigenvalues of for is found. We also prove that for large enough positive value of β the system exhibits persistent currents.

Singular configurations have long been out of consideration in the study of many-particle systems. In this paper we show that a group theoretical method can provide boundary conditions on wavefunctions at singular configurations of many particles. The consequences of this method are qualitatively stated as follows. (i) If the projection of angular momentum on an axis does not vanish, the particles will not be aligned on the axis, i.e. the wavefunction vanishes at collinear configurations along the axis. (ii) If the total angular momentum does not vanish, the simultaneous collision at a point will not take place, i.e. the wavefunction vanishes at the configuration of simultaneous collision. Furthermore, on the assumption of analyticity of the wavefunction at singular configurations, the behaviour of the wavefunction around the singular configurations shows that (a) the larger the projection of angular momentum on an axis becomes, the less the particles are likely to be aligned on the line, and that (b) the more the total angular momentum grows, the less likely the particles are to collide at a point. The proof is carried out for three-body systems by transforming the power series expansion of a wavefunction into a Fourier series expansion in terms of the angular momentum eigenfunctions, in both cases of collinear configurations and of the triple collision, without reference to the Hamiltonian operator. These results are described quantitatively in terms of the angular momentum quantum numbers in propositions 1 and 3 in the text.

In part I (Kanemitsu S et al 2003 J. Northwest University) we have made explicit use of the Mellin–Barnes integrals to prove the Chowla–Selberg-type Bessel series expressions for zeta-functions associated with lattice structures. In this paper we shall make implicit use of Mellin–Barnes integrals, as embedded in our theory of modular relations and functional equations, to reveal relationships between the structure of Madelung constants of the NaCl and CsCl lattices. Namely, we shall elucidate the relation between the structures of the NaCl lattice and those of the CsCl lattice, so to speak using the symmetry of the zeta-function, i.e. using their functional equations. Thus we shall emphasize the symmetry properties of the zeta-functions, restoring the Schlömilch series and Hardy's theory of K-Bessel functions, to prove the functional equations, and then to prove the recurrence relations for the lattice zeta-functions.

The Vandermonde determinant plays a crucial role in the quantum Hall effect via Laughlin's wavefunction ansatz. Herein the properties of the square of the Vandermonde determinant as a symmetric function are explored in detail. Important properties satisfied by the coefficients arising in the expansion of the square of the Vandermonde determinant in terms of Schur functions are developed and generalized to q-dependent coefficients via the q-discriminant. Algorithms for the efficient calculation of the q-dependent coefficients as finite polynomials in q are developed. The properties, such as the factorization of the q-dependent coefficients, are exposed. Further light is shed upon the vanishing of certain expansion coefficients at q = 1. The q-generalization of the sum rule for the squares of the coefficients is derived. A number of compelling conjectures are stated.

The Marumori–Yamamura–Tokunga boson expansion is used to describe multiboson processes. The reliability of the time evolution predicted by generalized coherent states has been investigated numerically by comparing its dynamics with the exact one. A link between the generalized coherent states and deformed bosons is established.

With a view to increasing the scope of applications of supersymmetric quantum mechanics (SUSY QM), we formulate the same in a complex phase space. Within this framework, the concept of shape invariance is reinvestigated and an insight into the eigenvalue spectra of non-Hermitian Hamiltonians is sought. The results are applied to a variety of potentials. We claim that the shape invariance for these potentials in the complex phase space can be retrieved in terms of the prescriptions already proposed in the conventional SUSY QM, in that the transformation of potential parameters takes the form of a reflection in the parameter space. Interestingly, some of these features turn out to be the generalization of the concept of quasi-parity used recently in the context of SUSY QM of PT-symmetric potentials. Further, a correspondence of the present approach with other complex formulations of SUSY QM and also in two real dimensions is demonstrated.

We investigate the classical dynamics of a two-electron atom and ions H⁻, He, Li⁺, Be²⁺, ..., in the collinear eZe configuration. We consider the case that the masses of all particles are finite. It is revealed that the mass ratio ξ between the nucleus and electron plays an important role for the dynamical behaviour of these systems. With the aid of an analytical tool and numerical computation, it is shown that, thanks to a large mass ratio ξ, the classical dynamics of these systems is fully chaotic, probably hyperbolic. Experimental evidence of this finding is also proposed.

This paper focuses upon the derivation of the non-classical symmetries of Bluman and Cole as they apply to Richard's equation for water flow in an unsaturated uniform soil. It is shown that the determining equations for the non-classical case lead to four highly non-linear equations which have been solved in five particular cases. In each case the corresponding similarity ansatz has been derived and Richard's equation is reduced to an ordinary differential equation. Explicit solutions are produced when possible. Richard's equation is also expressed as a potential system and in reviewing the classical Lie solutions a new symmetry is derived together with its similarity ansatz. Determining equations are then produced for the potential system using the non-classical algorithm. This results in an under-determined set of equations and an example symmetry that reveals a missing classical case is presented. An example of a classical and a non-classical symmetry reduction applied to the infiltration of moisture in soil is presented. The condition for surface invariance is used to demonstrate the equivalence of a classical Lie and a potential symmetry.

The (2+1)-dimensional Eckhaus-type extension of the dispersive long wave (EEDLW) equation is investigated, which was obtained in the appropriate approximation from the basic equations of hydrodynamics. Though it has no Painlevé property, we gain an auto-Bäcklund transformation (aBT) by truncating the Laurent series expansion at O(w⁰). In particular, the special one of the aBT establishes a relationship between the EEDLW equation and a set of three linear partial differential equations involving the well-known heat equation. Finally many types of new exact solutions of the EEDLW equation are found from the obtained aBT and some proper ansätze, which may be useful to explain some physical phenomena.

Solitons, negatons, positons, rational-like solutions and mixed solutions of a non-isospectral equation, the Korteweg–de Vries equation with loss and non-uniformity terms, are obtained through the Wronskian technique. The non-isospectral characteristics of the motion behaviours of some solutions are described with some figures made by using Mathematica. We also derive an infinite number of conservation laws of the equation.

Based on extraction of resonances from quantal time delay, a theorem relating quantal time delay and the number of resonances below a certain energy is proved here. Several illustrations from quantum mechanics, neutron reflectometry and hadron resonances are presented.

An su(1, 1) dynamical algebra to describe the continuum part of the spectrum for the modified Pöschl–Teller potential is proposed. The space associated with this algebra is given in terms of a family of orthonormal functions {Φ^σ_n} characterized by the parameter σ. This set is constructed from polynomials which are orthogonal with respect to a weighting function corresponding to a Pöschl–Teller ground state. An analysis of the associated algebra is investigated in detail. The functions are identified with Pöschl–Teller-like functions associated with different potential depths. We prove that by choosing appropriately the parameter σ it is possible to decouple the discrete states of a given parity from the continuum part of the spectrum. A discussion of the matrix elements of the dipole function operator is also included.

We revisit the application of different separability criteria by recourse to an exhaustive Monte Carlo exploration involving the pertinent state space of pure and mixed states. The corresponding chain of implications of different criteria is in such a way numerically elucidated. We also quantify, for a bipartite system of arbitrary dimension, the proportion of states ρ that can be distilled according to a definite criterion. Our work can be regarded as a complement to the recent review paper by Terhal B (2002 Theor. Comput. Sci.287 313). Some questions posed there receive an answer here.

We present an implementation of Grover's algorithm in the framework of Feynman's cursor model of a quantum computer. The cursor degrees of freedom act as a quantum clocking mechanism, and allow Grover's algorithm to be performed using a single, time-independent Hamiltonian. We examine issues of locality and resource usage in implementing such a Hamiltonian. In the familiar language of Heisenberg spin–spin coupling, the clocking mechanism appears as an excitation of a basically linear chain of spins, with occasional controlled jumps that allow for motion on a planar graph: in this sense our model implements the idea of 'timing' a quantum algorithm using a continuous-time random walk. In this context we examine some consequences of the entanglement between the states of the input/output register and the states of the quantum clock.

The theory of scale relativity provides a new insight into the origin of fundamental laws in physics. Its application to microphysics allows us to recover quantum mechanics as mechanics on a non-differentiable (fractal) spacetime. The Schrödinger and Klein–Gordon equations are demonstrated as geodesic equations in this framework. A development of the intrinsic properties of this theory, using the mathematical tool of Hamilton's bi-quaternions, leads us to a derivation of the Dirac equation within the scale-relativity paradigm. The complex form of the wavefunction in the Schrödinger and Klein–Gordon equations follows from the non-differentiability of the geometry, since it involves a breaking of the invariance under the reflection symmetry on the (proper) time differential element (ds ↔ −ds). This mechanism is generalized for obtaining the bi-quaternionic nature of the Dirac spinor by adding a further symmetry breaking due to non-differentiability, namely the differential coordinate reflection symmetry (dx^μ ↔ −dx^μ) and by requiring invariance under the parity and time inversion. The Pauli equation is recovered as a non-motion-relativistic approximation of the Dirac equation.

The new central and noncentral potential functions (CPFs and NCPFs) of a molecule depending on the coordinates of the nuclei are introduced. Using complete orthonormal sets of Ψ^α-exponential-type orbitals (Ψ^α-ETOs) introduced by the author, the series expansion formulae for the multicentre electronic attraction (EA), electric field (EF) and electric field gradient (EFG) integrals over Slater-type orbitals (STOs) in terms of CPFs and NCPFs are derived. The relationships obtained are valid for the arbitrary location, quantum numbers and screening constants of STOs.

The study of the density of independent harmonically confined Fermions is of considerable current interest because of (a) quantum dots and (b) the experiments of DeMarco and Jin on some alkali metal vapours in magnetic traps. Our concern here is with relativistic effects, first in harmonically confined assemblies of electrons as in (a) above and secondly with a stronger type of Fermion confinement typified again by electrons but now in a quartic rather than quadratic potential. While our numerical illustrations are for dimensionalities d = 1 and 3, some analytical results are presented in d dimensions.

We establish systematic consolidation of the Aharonov–Bohm (AB) and Aharonov–Casher effects including their scalar counterparts. Their formal correspondences in acquiring topological phases are revealed on the basis of the gauge symmetry in non-simply-connected spaces and the adiabatic condition for the state of magnetic dipoles. In addition, investigation of basic two-body interactions between an electric charge and a magnetic dipole clarifies their appropriate relative motions and discloses physical interrelations between the effects. Based on the two-body interaction, we also construct an exact microscopic description of the AB effect, where all the elements are treated on equal footing, i.e., magnetic dipoles are described quantum-mechanically and electromagnetic fields are quantized. This microscopic analysis not only confirms the conventional (semiclassical) results and the topological nature but also allows one to explore the fluctuation effects due to the precession of the magnetic dipoles with the adiabatic condition relaxed.

Time-independent perturbation theory in the scattering domain, developed on the basis of the quantum probability equation (Milward G C and Wilkin C 2001 J. Phys. A: Math. Gen.34 5101), is extended to study the energy dependence of the S-wave scattering phase shifts and probability density to second order in the energy change. At low energies, the Bethe formula for the effective range is reproduced and an analogous formula for the shape-dependent term is also derived. The energy dependence of the probability density at short distances is important in final state interaction theory and the method allows the investigation of some of the prescriptions used there. Results from the present approach are illustrated with soluble models.

Observables of quantum or classical mechanics form algebras called quantum or classical Hamilton algebras (HAs), respectively (Grgin and Petersen 1974 J. Math. Phys.15 764, Sahoo 1977 Pramana8 545). We show that the tensor product of two quantum HAs, each characterized by a different Planck's constant (PC), is an algebra of the same type characterized by yet another PC. The algebraic structure of mixed quantum and classical systems is then analysed by taking the limit of vanishing PC in one of the component algebras. This approach provides new insight into failures of various formalisms dealing with mixed quantum–classical systems. It shows that in the interacting mixed quantum–classical description, there can be no back-reaction of the quantum system on the classical system. A natural algebraic requirement involving restriction of the tensor product of two quantum HAs to their components proves that PC is unique.

We study a model loosely inspired by the giant carbon molecule C₆₀, namely a negligibly thin spherical shell of radius R, carrying a continuous fluid with mass and charge surface-densities m/a² and e/a² (plus an inert overall-neutralizing charge distribution), so that a mimics some mean inter-electron spacing, comparable say to the Bohr radius. The Casimir energy B is the total zero-point energy of the exact multipolar normal modes, minus that of empty space, minus the self-energy of the given amount of material at infinite dilution. Subject to a Debye-type cutoff l ⩽ L on angular momenta l, but needing no frequency cutoff, B is a well-defined function of R, x ≡ e²/mc²a, and X ≡ R/a, expressible in terms of the multipolar phase shifts δ^TE,TM_l. We consider it only for X ≫ 1 ⇒ L ∼ X. Realistically one has x ≪ 1, but μ ≡ 4πxX can be large or small. Then B is always dominated by terms of order stemming from TM modes; but the pattern of corrections as functions of x and X is intricate, and accessible only through the Debye (uniform) expansions of the Bessel functions figuring in the δ_l. Historical interest attaches to Boyer components, far-subdominant parts of B having the form (ℏc/R)C_B, where C_B is a pure number. When μ ≪ 1 (as in C₆₀) there are none, because all corrections are at least of order x^1/2, and none are proportional to 1/R. But a Boyer component does exist when μ ≫ 1 (as in macroscopic shells), with C_B = [3/64] − [(9/4096)(π²/8 − 1)] + ⋅ ⋅ ⋅ ≃ 0.0464. The two terms come from orders 1 and 2 of the Debye expansion; the contribution from order 0 vanishes because of an apparently fortuituous cancellation between TE and TM. The most precise value proposed so far is C_B = 0.046 1765; but the significance of comparisons is unclear, because previous calculations mistakenly treat the Boyer component as if it included all of B in a hypothetical perfect-reflector limit x → ∞.

It is shown that non-small angle multiple elastic scattering in matter leads to much stronger suppression of bremsstrahlung (BS) by high-energy particles in the long wave range of emission spectrum compared with the quenching predicted by Landau Ł D and Pomeranchuk I Ya (1953 Dokl. Akad. Nauk SSSR92 535, 735) and Migdal A B (1954 Dokl. Akad. Nauk SSSR96 49, 1956 Phys. Rev.103 1811). This manifests itself as the rearrangement of the BS spectrum of soft photons in the far long wave region.

We exploit the geometrical superfield formalism to derive the local, covariant and continuous Becchi–Rouet–Stora–Tyutin (BRST) symmetry transformations and the non-local, non-covariant and continuous dual-BRST symmetry transformations for the free Abelian 1-form gauge theory in four (3 + 1)-dimensions (4D) of spacetime. Our discussion is carried out in the framework of BRST invariant Lagrangian density for the above 4D theory in the Feynman gauge. The geometrical origin and interpretation for the (dual-)BRST charges (and the transformations they generate) are provided in the language of translations of some superfields along the Grassmannian directions of the six (4 + 2)-dimensional supermanifold parametrized by the four spacetime and two Grassmannian variables.

A self-action problem for a point-like charged particle arbitrarily moving in flat spacetime of six dimensions is considered. A consistent regularization procedure is proposed which relies on energy–momentum and angular momentum balance equations. The structure of the angular momentum tensor carried by the retarded 'Liénard–Wiechert' field testifies that a point-like source in six dimensions possesses an internal angular momentum. Its magnitude is proportional to the square of acceleration. It is the so-called rigid relativistic particle; its motion is determined by the higher derivative Lagrangian depending on the curvature of the worldline. It is shown that the action functional contains, apart from the usual 'bare' mass, an additional renormalization constant which corresponds to the magnitude of the 'bare' internal angular momentum of the particle.

A new and short proof of expression (3.2e) of Ališauskas (2002 J. Phys. A: Math. Gen.35 7323) with obvious triangle conditions for integrals of the Jacobi polynomials is proposed, without any allusion to the special isofactors of Sp(4). Expressions (3.6c) and (3.12) are corrected.