Table of contents

The discussion of boson realisations of symplectic algebras requires as an essential ingredient an operator K needed for the passage from the Dyson to the Holstein-Primakoff realisation. In previous papers the matrix form of K² was derived through appropriate recursion relations. In the present analysis the authors show that K² can be determined in explicit and closed form as the overlap of coherent states of the symplectic group. The matrix elements of K² can then be obtained by expanding this overlap in terms of appropriate eigenstates.

The Coulomb S-matrix is calculated via spherical functions on a hyperboloid. Relations with the standard results are discussed, and further prospects of this viewpoint are noted.

The authors explicitly determine the band spectrum of a periodically strongly singular Schrodinger operator in L²(R) associated with the differential expression -d²/dx²+(s²-1/4)/cos² x+(s'²-1/4)/sin² x, x epsilon R,x epsilon pi Z/2, 0<s,s'<1. The corresponding density of states is calculated analytically.

The author recasts a generalised Schrodinger-Langevin equation into a set of hydro-dynamical equations and draws some analogies with a classical (plasma) fluid theory describing the motion of charged particles in a neutralised background.

The relation between response function and correlation function is derived for the stochastic process associated to the ground state within the framework of stochastic mechanics. The new relationship clarifies the energy-conserving character of Nelson's stochastic processes.

Cluster-cluster aggregation is studied via a computer simulation in the chemically limited regime. True polydisperse flocculation is compared with idealised monodisperse aggregation in two and three space dimensions in terms of fractal dimensions and the exponents associated with the scaling of the reaction rates with cluster size.

The authors introduce and discuss a new model in which the growth sites are driven with a power law, r^alpha , away from the original seed. The concept of 'forgotten' growth sites is developed. They find that the fraction of forgotten growth sites, the 'order parameter' for this problem, is zero for alpha below a critical value alpha _c=1, and increases sharply for alpha > alpha _c with a critical exponent beta approximately=²/₃ in two dimensions.

Random walk models with disordered waiting times (site disorder) are analysed using a real space renormalisation group method. Anomalous diffusion is found in all dimensions whenever the first moment of the distribution of waiting times diverges. Above two dimensions the fixed points are described by continuous time random walks with non-analytic waiting time functions. Below two dimensions the fixed points are themselves disordered random walks where the disorder is characterised by a stable probability distribution. A fractal dimension is defined for site disordered lattices and, below two dimensions, a hyperscaling law is found which relates the fractal dimension, the Euclidean dimension, and the exponent describing the displacement of the walker.

The authors show that the ensemble of scattering matrices defined by the Poisson kernel in the statistical theory of nuclear reactions, is uniquely determined under the analyticity-ergodicity requirement and maximisation of Shannon's entropy.

The semi-infinite generalised Blume-Emery-Griffiths model of ³He-⁴He mixtures with a vector order parameter is analysed by means of Migdal-Kadanoff recursion relations. Superfluid film formation near the wall is put into evidence as a Kosterlitz-Thouless type transition. At higher ³H concentrations one observes the separation of a ⁴He rich normal film near the wall.

The corrections to scaling at and above the roughening transition T_R in the restricted solid-on-solid model are studied by means of finite-size scaling for semi-infinite strips of width 2 to 10. It is shown that at a temperature T_SD not far above the roughening temperature, where the restricted solid-on-solid model is invariant under the duality transformation of the five-state clock model, the fugacity of the sine-Gordon operator must vanish. For the dual truncated planar model this means that the core energy of the vortices diverges. Between T_S and T_R the corrections to scaling do not reach their asymptotic behaviour, but are dominated by the contributions from the ( Delta phi )⁴ operator associated with the restriction to steps of height one.

The authors discuss the mirror theory of spin systems with a surface. In the semi-infinite system, the 2-point correlation function at bulk critical temperature depends only on the 'real' distance and the 'image' distance, as was shown by the present authors within the framework of 1/n expansion and more recently by Cardy with the use of the conformal invariance. According to this mirror theory, they directly show the scaling relation 2 eta _{perpendicular to} - eta /sub /// eta . They also find a universal combination of amplitudes. They check the mirror theory by means of the epsilon (=4-d) expansion and present the explicit form of the correlation function in real space up to O( epsilon ²). The resulting surface critical exponents eta _{perpendicular to} and eta /sub /// coincide with those obtained previously.

The authors give an algebraic characterisation of the dynamical systems which preserve the coherence of a generalised coherent state defined for a Lie group. The breaking of coherence is related to singularities appearing in the S matrix. They show a regularisation procedure to eliminate such singularities based on the jet realisation of the diffeomorphism group induced by contact transformations on the state manifold.

A formalism of quantum kinematics which rests on the regular representation of a Lie group is proposed. This representation leads to explicit canonical commutation relations for non-Abelian dynamical variables which, together with the Lie algebra, define the kinematic algebra. Generalised equations of motion for the group-parameter-dependent operators, as well as generalised wave equations, are introduced over the group manifold considered as a background arena. The formalism proposed affords a method of geometric quantisation stemming directly from the observed symmetries of a system.

The quasi-periodic solutions for the Boomeron equation are determined by means of function-theoretical methods related to Riemann surfaces and theta functions. Also determined are the so-called Boomerons as degenerations of the quasi-periodic solutions. Moreover it is indicated that there are no higher-order Boomeron equations than the second-order one.

An efficient and accurate algorithm for Monte Carlo simulation of a solution to the diffusion equation with drift and branching terms is derived by expanding the Green function operator in time.

Numerical values for V( rho , z) in the vicinity of a parallel plate capacitor have been calculated using both the Love integral-equation method and a relaxation method. The two methods agree. Figures are presented showing equipotentials for three different values of kappa , the ratio between plate separation and radius.

The structure and implementation of a new general iterative method for diagonalising large matrices (the 'residual minimisation/direct inversion in the iterative subspace' method of Bendt and Zunger) are described and contrasted with other more commonly used iterative techniques. The method requires the direct diagonalisation of only a small submatrix, does not require the storage of the large matrix and provides eigensolutions to within a prescribed precision in a rapidly convergent iterative procedure. Numerical results for two rather different matrices (a real 50*50 non-diagonally dominant matrix and a complex Hermitian 181*181 matrix corresponding to the pseudopotential band structure of a semiconductor in a plane wave basis set) are used to compare the new method with the competing methods. the new method converges quickly and should be the most efficient for very large matrices in terms both of computation time and central storage requirements; it is quite insensitive to the properties of the matrices used. This technique makes possible efficient solution of a variety of quantum mechanical matrix problems where large basis set expansions are required.

A particle moves in circular arcs with Larmor radius R between specular reflections at the smooth convex boundary of a planar region. The dynamics depends on the value of R in relation to the extreme curvature radii rho _min and rho _max and the radius R* of the largest circle that can be inscribed in the boundary. For R<R* some orbits are complete Larmor circles and constitute an integrable component of the motion; all other orbits bounce repeatedly. For rho _min<R< rho _max there are 'flyaway intervals' on the boundary for which glancing orbits are a powerful source of chaos in the map (on the phase cylinder) relating successive bounces; this type of chaos is a characteristic feature of magnetic billiards. For sufficiently large R the simplest closed orbits consist of two arcs associated with diameters of the boundary; their existence and stability can be determined. In several regimes where motion consists of short skips between nearby boundary points (including the strong-field case R to 0), an explicit adiabatic invariant can be found which gives an excellent approximation to the exact invariant curves in these regimes. Computations for a magnetic billiard with elliptic boundary illustrate the theory.

Earlier works on screened Coulomb potentials using Rayleigh-Schrodinger perturbation theory have been re-examined. Instead of working with the usual Hulthen potential as the unperturbed Hamiltonian, the authors propose that a scaled Hulthen potential with modified strength and screening coefficient represents the lowest-order approximation for the static-screened Coulomb and exponential cosine-screened Coulomb potentials. The scale parameter appearing in the new Hulthen potential is then determined from the notion of the virial theorem and intuitive physical arguments. It is found that the accuracy of the predicted energy eigenvalues for the bound s states improves significantly even when the screening parameter is large and quite close to its critical values for which the quantum state becomes just bound. In spite of the simplicity of the approach, the numerical results compare fairly well with those obtained from rigorous analytic approximation methods.

A simple procedure for solving the Schrodinger equation is presented. It is based upon an iterative solution of the secular equation. A large enough convergence rate is obtained by using a basis set of properly scaled functions. The effect of the scaling parameter on the convergence rate is studied in order to improve the calculation. The method is applied to simple, though non-trivial, quantum mechanical models such as the quartic, sextic and octic anharmonic oscillators, a double well potential, and the linear confining potential. Highly accurate eigenvalues for all values of the coupling parameter are obtained.

Two series of Clebsch-Gordan type linearisation relations are derived for the most general product of the Laguerre polynomials, L_n1^{alpha 1}(u₁x)L_n2^{alpha 2}(u₂x), which differ in orders, n, weights, alpha , and scaling multipliers, u. The general form and particular cases of coefficients in the expansion of the polynomial x^kL_n1^{alpha 1}(u₁x) . . . L_nN^{alpha N}(u_Nx) in terms of the Laguerre polynomials are established. The applications to hydrogen-like functions and Morse oscillators are indicated. Connection with an earlier Carlitz expansion, the technical links with the hyperspherical harmonics formalism and different approaches to the important Koornwinder's positivity theorems are discussed briefly.

The two-dimensional one-component plasma near a metal wall is exactly solvable at a special value of the coupling constant. The author calculates the grand canonical partition function and correlation functions in strip geometry along the interface. By taking the strip width to infinity, the author obtains the exact statistical mechanics of a model of the metal-electrolyte boundary. Sum rules are discussed.

The calculation of ensemble averages in one-dimensional walks with weights given by exp(-g Sigma (n_i)^alpha ), where n_i is the number of times a site is visited, is simplified by reducing the problem to one of calculating binomial factors for partitions and their permutations. The combinatorial factors are generalised to the case of Cayley trees and all lattices without closed loops. In one dimension the combinatorial formulae are shown to reduce the known cases when alpha =0 and alpha =1. The authors also use these formulae to extend the exact enumeration results, for the one-dimensional generalised random walk, from 21 to 29 terms; and the series are analysed using Pade approximant techniques.

The author investigated the motion of many non-interacting particles with inertia in a viscous medium subject to an external, random force field that varied both in space and time. By varying parameters in the problem, it was found that there is a transition between two regimes. One regime, characterised by low mass or high viscosity, leads to the aggregation of particles. The other regime, high mass or low friction, leads to a complete lack of aggregation. The transition between these two regimes is marked by a discontinuous change in the order parameters defined for this problem. The relation between this work and previous work on iterated functions is discussed.

The properties of a simple one-dimensional iterated map in the presence of noise are examined. It is demonstrated that there are three distinct classes of noise that generate three 'phases' of the iterated function. Each phase shows universal properties that are independent of the specific form of noise, if this noise is chosen form a given class. The physical interpretation of the model in terms of aggregation is discussed.

Using the phenomenological renormalisation (PR) method the authors obtain the critical probability for partially directed site percolation on the square and triangular lattices (p_c=0.6317+or-0.006 and p_c=0.5468+or-0.005, respectively). They discuss the corrections to scaling in the calculation of the anisotropy exponent theta (= nu /sub //// nu _{perpendicular to} ) and the critical probability, and show that it is necessary to work with at least two correction terms. This is in contrast to the usual analysis where only one correction exponent is used in PR calculations.

Evidence is presented in favour of the hypothesis that the Zamolodchikov model (an exactly solvable three-dimensional lattice model without any temperature-like parameters) is critical. The evidence is obtained by generalising the Zamolodchikov model to include a temperature-like variable. The magnetisation curve of this model is then studied using a modified form of a variational approximation formulated earlier. Also the authors show the two-layer Zamolodchikov model corresponds to a critical free-fermion model.

A straightforward algebraic technique is introduced in order to establish duality relations between spin-1/2 lattice systems. A new self-duality relation is found.

By using a procedure introduced in the previous work, the author considers duality transformations for spin-1/2 lattice systems. The author shows that, in some cases, given a self-dual model, a new self-dual model can be obtained from it. Several examples of this fact are given. In particular the author analyses a system consisting of two interacting Baxter-Wu models, which present close analogies with the Ashkin-Teller model.

High-precision Monte Carlo data are used to estimate the exponents which govern the asymptotic behaviour of the recently introduced indefinitely-growing self-avoiding walk in two dimensions. For this walk the exponent gamma is by definition equal to one. Applying the same methods which are used to extract the exponents from exact series enumeration, the authors give an estimate for the exponent nu of 0.567+or-0.003. The leading corrections to this asymptotic behaviour are also calculated.

The authors determine equilibrium crystal shapes for a triangular Ising model with isotropic nearest-neighbour interactions. At zero temperature the shape is an equilateral hexagon for ferromagnetic interactions, and an equilateral triangle for antiferromagnetic interactions in the presence of a weak field, H<or=2J. When the magnetic field is increased beyond 2J the shape varies continuously with the field, in the form of a hexagon with three-fold symmetry. For still higher fields, H>or=4J, the shape is again an equilateral triangle, now oriented pi /3 relative to low-field crystals. Finite-temperature shapes are determined by means of a solid-on-solid calculation of the orientation-dependent interfacial free energy.

Recent relations that relate the exponent s of superconducting percolating networks to other percolation exponents are discussed. The author analyses the available exact and numerical results for s to gain insight into the possible relation between s and the geometrical exponents of percolation and the structure of superconducting percolation networks. The results are given geometrical interpretation. The author also discusses the random walk statistics of the 'termite' which executes a random walk on the superconducting percolation networks. In particular the author proposes an expression for the mean number of distinct sites visited by the termite and interpret it in terms of the statistics of a random walk with a transition probability whose variance is infinite.

The Kelvin equation for the equilibrium vapour pressure outside a drop of liquid is widely quoted but its derivation is subject to three assumptions which are not obviously justified. Consequently some doubt has recently been thrown on the value of the equation for estimating the true vapour pressure. It is shown in general, and explicitly and in detail for a Lennard-Jones fluid, that the equation is valid to a few per cent even for temperatures approaching the critical temperature and for microscopic drops insofar as homogeneous thermodynamics is valid. It is noted that the spinoidal radius of a drop is numerically equal to the surface thickness for a Lennard-Jones fluid. This relation may well be general and useful.

The Schrodinger equation for an atom having N electrons (N=1,2,3, . . . ) has been solved exactly within the framework of generalised power series. The members of this series can easily be calculated from a two-term recursion. The present formulation is suitable both for computational as well as for analytical work.

In the light of renewed discussions about variations in the fundamental constants the authors have investigated the effect of changing the fine-structure constant alpha on the chemistry of medium and heavy elements important for metal-containing biological molecules. Relativistic Hartree-Fock calculations performed for Fe²⁺ and Fe³⁺ ions (important, e.g., for haemoglobin) indicate that a five-times larger alpha can change energy levels and valence directionality significantly. Other relativistic effects in chemistry are discussed.

A three-dimensional system consisting of a set of interacting two-dimensional Ising models is introduced. It is shown that this model is self-dual. It contains the three-dimensional Ising model as a particular case.