Table of contents

In the O(4,2) group an algebraic scheme of perturbation theory is proposed which enables one to find the energy and wavefunctions of bound states of a hydrogen-like atom in the field of a point charge placed at large distance R. It is shown that in each order of R^-1 the correction to the wavefunction is expressed through a finite number of Coulomb functions of the discrete spectrum with a modified charge. Simple analytical expressions are obtained for the first and second corrections.

It is shown that the methods of supersymmetric quantum mechanics can be used to obtain the complete energy spectrum and eigenfunctions of the Dirac equation for an attractive Coulomb potential.

The authors have considered in the framework of the field-theoretical approach the asymptotics of random walks in media with quenched disorder which is described by drift velocity field V(x) subjected to various constraints. The disorder has proved to be relevant in d<or=2 dimensions leading (at d=2) to superdiffusion behaviour for the solenoidal random drift ( delta V_i/ delta x_i=0) and to subdiffusion behaviour for the potential random drifts ( delta V_i/ delta x_j- delta V_j/ delta x_i=0).

The authors show that the 'true' self-avoiding walk with long-range interactions leads to a renormalisable field theory. They compute the characteristic exponent of the gyration radius up to first order in the difference between the upper critical dimension and space dimensionality, and compare it with the estimates based on Flory-like arguments.

The configuration-averaged vibrational spectrum of Eden clusters with N sites on a d-dimensional hypercubical lattice is determined exactly in the limit d>>log N>>1. It is shown that the spectral dimension of these clusters is 2.

The probability density of the displacement or end-to-end distance of a random walk on the incipient infinite percolation cluster in d=2 dimensions is studied by an exact enumeration method. The numerical data suggest specific forms for the probability density both in the chemical distance variable l and the geometric distance r.

Employs exact enumeration methods to study a number of configurational properties of self-avoiding random surfaces embedded in a three-dimensional simple cubic lattice. Self-avoiding surfaces are defined as a connected set of plaquettes in which no more than two plaquettes may meet along a common edge, and in which no plaquette can be occupied more than once. Based on enumerating surfaces containing up to 10 plaquettes, the author finds: (a) the number of n-plaquette surfaces, c_n, varies as mu ⁿn^{gamma -1}, with mu =13.2+or-0.2 and gamma =0.22+or-0.06, (b) the average number of perimeter edges of n-plaquette surfaces, (p_n), varies linearly with n, and (c) the mean-square radius of gyration of n-plaquette surfaces, (R_g²(n)), varies as n^{2 nu} , with 2 nu =1.075+or-0.05.

Two models of diffusion in superconducting percolation networks are studied on a one-dimensional system. The authors use chains of 5*10⁵ sites and study the scaling properties of the mean-squared displacement of 'termites' which execute random walks on the network, mean number of distinct sites visited and mean number of visits to the origin. They find that although these quantities can be described by well defined scaling laws, the values of the critical exponents appear to depend on the time interval in which the walks are studied. They observe at least four different scaling regimes. The implications of these results for higher-dimensional systems are discussed.

The one-dimensional Ising model with oscillating interaction is considered in the presence of a magnetic field. The devil's staircase found for convex interaction is destroyed and the system crosses over to a harmless staircase.

Examines and evaluates the functional determinants associated with a class of periodic operators which arise in the quantum theory of solitons. The functional determinants are expressed in terms of a finite subset of the spectrum of the operator, a subset which corresponds to the edges of the valency or conduction bands of the Schrodinger equation.

A new method is described for the automatic solution of linear ordinary differential equations by the use of Taylor series. This new method is shown to be superior in speed and accuracy to conventional methods. The method is illustrated for Coulomb wavefunctions, confluent hypergeometric functions, zeros of Bessel functions and s-wave phaseshift for e-H scattering in the static approximation.

A necessary and sufficient condition for the equivalence of the conformal invariance of the Euler equation and the divergence invariance of the associated variational integral is obtained in terms of the generators of the local Lie group of transformations under which invariance is studied.

The authors discuss the characterisation of intermittency in chaotic dynamical systems by means of the time fluctuations of the response to a slight perturbation on the trajectory. A set of exponents is introduced which generalise the maximum Lyapunov characteristic exponent. The link with the statistical mechanics formalism is emphasised and they show that the exponents are connected to a free energy formally defined for chaotic systems. They perform some analytical computations in simple cases and give a few numerical examples.

The authors give a group-theoretic construction for a one-parameter family of alternative Lagrangians for the problem of a particle moving under a spherically symmetry potential, recently discovered by Henneaux and Shepley (1982). The authors also discuss the question of what distinguishes the standard Lagrangian from the others, which is of some importance in relation to the quantisation of the hydrogen atom, for example.

Prolongation structures are determined for the matrix Korteweg-de Vries and the Boomeron equation by using the integrability condition for a linear system of first-order equations. Symmetries of these prolongation structures are used to derive Backlund transformations and to construct solutions for both equations.

Iterative averaging, which has previously been shown to be equivalent to the renormalisation group, was applied directly to the Navier-Stokes equations in k space. The better frequency separation achieved by this technique (when compared to convention RG filtering and averaging operations) was shown to be due to the order in which operations were carried out, rather than to any difference in underlying assumptions. An expression for the eddy viscosity for wavenumber scales k<or=k_c (say) was obtained in terms of the energy spectrum for k>or=k_c. For k<<k_c, the eddy viscosity became constant. For k to k_c the eddy viscosity had a gentle roll-off, which may be compared with the cusp at k=k_c found with renormalized perturbation theory.

Let H₀(B) denote the Hamiltonian of a free electron in a magnetic field B. Let V be a periodic potential. The authors show that if an interval (a,b) is not in the spectrum of H₀(B₀)+V for some B₀, then it is not in the spectrum for all B sufficiently close to B₀.

The off-diagonal hypervirial theorems and a matrix product rule are applied to the perturbed one-dimensional harmonic oscillator in a new method of obtaining perturbation series for the off-diagonal elements (m mod x mod n). The author gives the Rayleigh-Schrodinger series for (0 mod x mod 1) to eighth order for a quartic perturbation.

States of two-body systems bound by a spherically symmetric potential whose Laplacian is positive are considered. Constraints are derived on the kinetic energies and radii of states with the lowest energy for a given angular momentum, which improve on previous results of the author. These constraints become relatively very tight at high angular momentum. Inequalities are also derived between the energies of states of differing angular momentum and particle mass.

The authors compare semiclassical and quantum mechanical (numerical) results for spectral statistics in the transition from order to chaos in the corresponding classical system. Scale invariant two-dimensional Hamiltonian are discussed and significant discrepancies are found only for level spacings smaller than one average spacing. For the semiclassical limit they use the result of Berry and Robnik (1984) for the nearest-neighbour spacing distribution and the extension of it to the Delta ₃ statistic.

Shows that when a point-like, neutral, but polarisable body moves with Cerenkov velocity in a medium whose collective excitations are in the ground state, there is a radiation process.

The mechanism of stochastic resonance is studied in the case of the Landau-Ginzburg equation stochastically and periodically perturbed, by taking advantage of recent developments on the stochastic partial differential equations. Analytical expressions are given for computing the exit times of the system and to estimate the range of the noise for which the stochastic resonance is possible.

The duality transformation is given for the gauge invariant random Potts model on a square lattice. Gauge invariance and duality properties of the partition function are used to demonstrate the existence of a frustration-dissociation phase transition in the ground state at a critical concentration of ferromagnetic bonds. The phase diagram is obtained with the aid of an annealed approximation.

The Kasteleyn-Fortuin relation (1969) is used to give a simple proof for the s-state Potts model of the relation beta sigma xi =1 between the correlation length in a particular direction on a planar lattice and the surface tension of an interface with corresponding orientation on the dual, thus generalising a well known result for the Ising model. The relation is also used to give an interpretation of the surface tension in the percolation problem in general dimensions.

The exact expressions of the disorder varieties and the corresponding values taken by the partition functions of the checkerboard Ising and Potts models with a magnetic field are obtained. Different specialisations and extensions of these exact results are examined.

The surface of the Eden model is investigated numerically by finite-size scaling, using a strip geometry. Three different versions are studied and it shows that the one mostly used previously exhibits strong finite-size corrections. In two dimensions it is found that the surface thickness simply grows as the square root of the width of the strip for infinitely long strips. This result, as well as the results in d=3 and d=4, suggest that the surface of the Eden shares some properties with the equilibrium models used to describe the roughening transition. Moreover, it is found that the surface thickness scales differently with the height of the cluster for infinitely large strips.

A master equation formulation of the kinetic, q-state Potts model is presented. It is shown that this formulation reduces to Glauber's dynamics in the Ising limit. The single-spin problem is considered and the properties of its q-1, temperature dependent relaxation times are studied. The linear chain problem is much harder than in the Glauber case-a variational method is used to obtain a lower bound for the dynamical critical exponent (z>or=3 for q>2). A discussion of these and related problems is presented.

A dynamical theory of spin glasses based on the (assumed) morphology of the ground state and on some scaling hypotheses is presented. It yields a closed formula for the AC susceptibility which can reproduce the experimentally observed behaviour of chi in both frequency and temperature. The time-dependent susceptibility is also analysed: it is found that equilibrium is reached in an extremely slow manner, and might never be observed in real experiments.

Generalised spin-1/2 and Potts quantum chains with m- (n-) site interactions for the z(x) component of the site variables are shown to be self-dual. Through duality, with m arbitrary, n=2 versions of the models are transformed into m independent Ising or Potts chains in a transverse field, allowing one to get the exact values of the critical exponents for any m.

It has recently been demonstrated by Skjeltorp (1983, 1984) that a monolayer of non-magnetic inclusions in a paramagnetic fluid crystallises when a perpendicular magnetic field is applied. Temperature is scaled by this field and 2D melting has been observed. Here the authors examine the energetics of the stability of the regular structures formed. Competing dipolar interactions and a frustration under distortions give a range of possibilities including a strictly 2D triangular lattice, a quasi 2D distorted triangular lattice and a quasi 2D square lattice. A second melting possibility becomes evident, but they restrict the analysis of the interactions to zero temperature. The underlying lattice energies are all close and the delicacy involved in calculating their energies, arising from the long-range character of the forces involved, requires a modification of the Ewald technique. They perform a stability analysis to find instability modes and their amplitudes.

The authors use a finite-size scaling calculation to study the phase diagram of the two-dimensional, three-state, chiral clock model. Bulk properties, the ferromagnetic transition temperature and associated exponents are calculated from the scaling of the susceptibility and interface behaviour, the position of the interface wetting line and its exponents, from considering the net adsorption. They also present preliminary results on interface properties which are sensitive to the transition between the paramagnetic and incommensurate phases.

A solution, exact to terms of second order in the temperature gradient, Del T, is obtained to a Fokker-Planck-type equation previously derived for the distribution g(v) in values v of the local heat flux in a large, homogeneous fluid phase. The solution assumes Del T is applied at time t=0, causing the heat flow to build up and relax toward a steady state. An exact expression for the steady value of (v²) is obtained to order ( Del T)² to test an earlier result, obtained by an approximate expansion of g. The exact solution is more useful than the earlier one, which involved truncation errors, for calculating correlation functions, relaxing terms in g are also calculated.

The asymptotic behaviour of the multi-soliton solution for the SU(N) sigma model is analysed. It is shown that if the diagonal matrices A₀ and B₀ in su(n) defining the vacuum solution are non-degenerate (their diagonal elements are all distinct), the multi-soliton solution is asymptotically diagonal.

Discusses the equations of motion and the conservation laws for a non-relativistic isospin carrying particle in a spherically symmetric monopole field and for a vanishing Yukawa coupling. In the 't Hooft-Polyakov monopole field the author finds no classical counterpart of the Rubakov effect. In the Prasad-Sommerfeld limit the author can solve the equations of motion analytically.

For lattice chiral and gauge models the authors develop an 1/N² expansion of the mean-field approximation. Special attention is paid to the free energy for which they also present as an 1/N² expansion the effect of fluctuations around the mean-field solution. The differences between U(N) and SU(N) are pointed out. Finally, for the chiral model they apply the mean-field saddle-point technique to compute the two-point correlation function.

The Lanczos method is an iterative scheme which when applied to the Hamiltonian H of a quantum system and starting from an initial vector (which is, in principle, an arbitrary vector of the associated Hilbert space) produces very good approximations to the eigenvalues at the extreme ends of the spectrum after a few iterations. In this comment the authors derive analytically a lower bound to the smallest eigenvalue and an upper bound to the largest eigenvalue of the Lanczos Hamiltonian of the system in terms of the expectation of H evaluated at the initial vector.

The method of large-N expansion has been applied to an attractive radial Gaussian potential to obtain its bound state energy levels. It has been shown that the shifted 1/N expansion technique provides a better approximation than the method of unshifted 1/N expansion and yields energy values which are in good agreement with the accurate numerical and analytic results.

Analyses the five-dimensional Ising model using Monte Carlo methods.

Uses the method developed by Achiam and Kosterlitz (1978) to obtain the dynamical critical exponent of the one-dimensional q-state Potts model. The author finds z=3 (q>2) and z=2 (q=2), thus confirming lower-bound estimates found previously.