Table of contents

The author considers the infinite products of d*d random matrices X_i which are taken according to a Bernoulli distribution (i.e. X_i = A with probability (1-p) and X_i=B with probability The maximum Lyapunov exponent is estimated by selecting a suitable periodic sequence. One need only then compute the eigenvalues of one particular matrix which is obtained as a finite product of matrices A and B. He also discusses the cases in which the method fails because of fluctuation effects.

Time-dependent harmonic perturbation in dynamical systems is frequently approximated by a sequence of infinitely short pulses (so-called kicks) of alternating sign. In order to improve this approximation the authors increase the number of kicks per perturbation period. The validity of this method is tested numerically on an exemplary classical system: a particle in the Rosen-Morse potential well, driven by an external harmonic perturbation.

By requiring a Hamiltonian system with several adjustable parameters to have a certain stability property, a family is created which includes as its members the integrable cases. The results, obtained analytically, for two Hamiltonian systems with two degrees of freedom, are presented.

The authors consider the critical behaviour of the spin-1 Hamiltonian of the Takhtajan-Babujian model. The Bethe ansatz equations for this antiferromagnet in a finite chain are investigated analytically and numerically. The conformal anomaly and critical exponents are obtained by exploiting their relations with the eigenspectrum of the finite system. The results strongly support the conjecture that the Wess-Zumino-Witten non-linear sigma model with topological charge k=2 is the underlying field theory for this statistical mechanics model.

The authors study the excited-stage energies of the s-wave hydrogen atom with the polynomial perturbation 2 lambda r+2 lambda ²r². They demonstrate the existence of a discontinuity in the eigenvalue spectrum at lambda =0 as lambda changes from positive to negative.

A method based on the Euler-Maclaurin formula is proposed to obtain all finite-size corrections to the energies of the ground and excited states for the one-dimensional Bose gas with delta function interactions. Scaling dimensions for all gapless states and some states having a macroscopic momentum are obtained.

It is shown that the operator content of frustrated sectors of c=1 Z_n invariant conformal field theories (including the n to $OI limit, the XXY model) can be obtained from the field theory of a free massless boson with appropriate boundary conditions. It is also shown that, for frustrated boundary conditions, the requirement of modular invariance is substituted by definite transformation properties of the partition function under the modular group, which are satisfied by the partition function constructed.

Using Monte Carlo method, the authors calculate the specific heat of branched polymers near a hard wall in the presence of a short-range attractive force between monomers and the wall. The specific heat exhibits a peak at a temperature T_m(N) depending on the size of the polymer N. Both T_m(N) and the height of the peak increase monotonically with N. The result agrees very well with a scaling function for the specific heat derived from a scaling analysis in analogy with linear polymers. In addition, the scaling analysis yields the adsorption temperature T_a approximately=0.583 and the crossover exponent phi approximately=0.714.

A gas of test particles in d dimensions, which interacts with a background host medium in the presence of external forces, is studied. Scattering, removal and self-generation collisions are taken into account. A generalised Nikolskii transform reduces the inhomogeneous Boltzmann equation for this system to the homogeneous one without background or external forces. The authors show that a background of field particles can confine the test gas, even in the absence of external forces.

It is proved that, in sufficiently high dimensions, the scaled self-avoiding random walk on the hypercubic lattice converges in distribution to Brownian motion. Convergence of the finite-dimensional distributions was shown elsewhere. In this paper tightness is shown.

A procedure, similar to the finite-size scaling, is proposed for systems with long-range interactions. It is applied to the one-dimensional Ising model with interactions decaying as 1/ mod i-j mod ^{1+ sigma} with distance mod i-j mod . The critical temperature and exponent nu are calculated for sigma <or=1.

The general epidemic process (GEP), a stochastic multiparticle process which exhibits a critical point near an absorbing state and leads to percolating clusters is studied in a finite environment. Using renormalisation group techniques, the authors calculate the linear relaxation time in a cubic geometry of finite size L, with periodic boundary conditions imposed. The corresponding scaling behaviour to O( epsilon )( epsilon =6-d, d being the spatial dimension) is presented in universal form.

The symmetry group of the generalised non-linear Schrodinger equation i psi _t+ Delta psi =a₀ psi +a₁ mod psi mod ² psi +a₂ mod psi mod ⁴ psi in three space dimensions is shown to be the extended Galilei group G(3), for a₁a₂ not=0, and the Galilei-similitude group G^d(3) (including a dilation) for a₁=oor a₂=0. All Lie subgroups of G(3) and G^d(3) are found. They will be used in a subsequent paper to obtain group invariant solutions of the equation.

A group theoretic method of obtaining the maximum number of non-vanishing independent constants required to describe a chosen physical property by the 18 polychromatic classes is presented. These numbers, in conjunction with the magnetic constants already obtained during the earlier study by this author (J. Phys. A: Math. Gen., 20, 47) in respect of these colour classes, may help identify the class of crystals that bear the possible geometric configuration envisaged by Indenbom et al. (1960). The effect of spins in crystals in their magnetic state on the non-magnetic physical properties is theoretically studied and a brief discussion of the results is presented.

A new approach to evaluating the static lattice energy of any given Wigner lattice is proposed. The method is much simpler than the traditional one and the results are much faster to evaluate. The new approach is applied to two-dimensional triangular lattices where it is shown that the triangular lattice is more stable than the square lattice. Three-dimensional hexagonal lattices are also investigated. In addition, the authors place many of their considerations on a rigorous mathematical footing.

It is shown that the recursion method, i.e. the continued-fraction expansion of the diagonal element of the resolvent, is equivalent to the exponential Toda lattice in the sense that the limiting identities in the former method have the same forms as the conservation laws in the latter lattice. The asymptotic continued-fraction coefficients (recursion coefficients) are then related to a particular motion in the Toda lattice. As a result of this relation, the analytic expressions for the simply oscillating asymptotic recursion coefficients in the two-band (single-gap) case are given explicitly in terms of the Jacobian elliptic functions and in the Fourier series forms.

The authors find some relationships between the usual AKNS scheme with the super one, when its elements take values from the Grassman algebra on a two-dimensional vector case. The solutions of these super AKNS equations are discussed.

Dynamics of a sine-Gordon kink and small-amplitude breather bound on an attractive localised inhomogeneity are studied. The rates of energy emission from the kink oscillating near the inhomogeneity and from a quiescent breather performing internal oscillations are calculated. In the semiclassical approximation, discrete quantised energy spectra for the solitons of both types are found, with regard to field-theory corrections to a kink's particle-like spectrum and to strong distortion of the breather's shape. The rates of radiative energy emission are interpreted as finite widths of the quantised quasiclassical energy levels. Oscillations of the breather as a whole near the inhomogeneity are also considered.

An analytic continuation procedure using a Taylor series is used to produce very accurate wavefunctions and eigenvalues for the one-dimensional anharmonic oscillator governed by the potential V x)=x²+λx²/(1+gx²).

A semiclassical analysis is given for general chaotic Hamiltonians of the short-time quantum dynamics of survival probabilities of pure initial states in canonical bases. This is illustrated for the Feingold-Peres coupled spin model which is classically chaotic. The dynamics predicted by the semiclassical arguments is compared with exact quantum mechanics and is found to be remarkably accurate even for as large as 0.23.

The authors use the shifted 1/N expansion method to calculate the energy eigenvalues of the non-polynomial oscillator V(r)=r²+λr²/(1+gr²). The results thus obtained are in excellent agreement with exact numerical results. It is also shown that the non-polynomial interaction can be made supersymmetric and in this case also the results obtained by the shifted 1/N method matches extremely well with the exact analytic results given by supersymmetry.

It is shown that the non-polynomial oscillator represented by V(x)=x²+ lambda x²//(1+gx²) can be given a supersymmetric form if the parameter lambda satisfies a certain constraint. In this case one finds an exact analytic expression for the ground state. Excited states corresponding to this potential are computed for several values of g (and lambda ) using the supersymmetric WKB method, the ordinary WKB method and also numerical integration. The results are compared.

In this paper high-order adiabatic approximate solutions of the Schrodinger equation for a quantum system with a slowly changing Hamiltonian are presented. The author not only obtains Berry's phase factor and strictly proves the quantum adiabatic theorem in the first-order approximation, but also discusses an observable effect of the second adiabatic approximation.

The author obtains a wide class of multidimensional polynomial potentials for which the Schrodinger equation allows the separation of variables in generalised ellipsoidal coordinates.

In a planar approximation to a Yukawa-type g psi * psi phi field theory with scalar fields psi and phi the author studies the Bethe-Salpeter (BS) equation for the scattering amplitude of the psi field in the case of vanishing psi wavefunction renormalisation constant Z₂=O. Due to the asymptotic behaviour of the non-canonical Psi propagator, given by the corresponding Dyson-Schwinger equation for Z₂=O, the Neumann series of the BS equation diverges for Euclidean values of the invariants and all masses m², mu ²>O. Being responsible for this divergence, only the asymptotic part of the propagator is subsequently retained in the BS equation. Using in the Euclidean metric an exactly soluble high-energy version of the BS equation and treating the difference as a perturbation, he derives a new but equivalent integral equation for the scattering amplitude. By contraction-mapping arguments he obtains existence and multiplicity results for solutions of this transformed equation. The asymptotic behaviour of these solutions is rigorously established and found to be oscillating.

The inverse scattering problem for the plasma wave equation is approached from the viewpoint of scattering theory for the reduced wave equation. It is shown that the scattered wave for the plasma wave equation has the large-t asymptotics which are expressed by the scattering amplitude for the Schrodinger operator associated with the plasma wave equation.

When the quantum planar rotor is put on a lattice its dynamics can be approximated by random walks on a circle. This allows for fast and accurate Monte Carlo simulations to determine the topological charge of different configurations of the system and thereby the theta dependence of the lowest energy levels.

The random tiling of the plane by a set of objects (e.g., rhombi), related by rotational (e.g. tenfold) symmetries, is a paradigm for the formation of quasiperiodic order due to entropy. Such tilings are mapped to a higher-dimensional space where they form hypersurfaces analogous to the interfaces in a solid-on-solid model. The author argues that the fluctuations of the hypersurface should be described by a gradient-squared free energy of entropic origin; this implies quasi-long-range order in d=2. He shows how the random tiling can be decomposed into layers, defines a transfer matrix, and give prescriptions for using this method to determine numerically the stiffness of the gradient free energy.

A complete description of all periodic low-temperature Gibbs states of ferromagnetic classical lattice gas models with a finite but arbitrary number of different particles is obtained.

Discusses the existence of knots in random self-avoiding walks on a lattice. Using Kesten's (1963) pattern theorem, it is shown that almost all sufficiently long self-avoiding walks on the three-dimensional simple cubic lattice contain a knot.

Kauffman's model is a randomly assembled network of Boolean automata. Each automaton receives inputs from at most K other automata. Its state at discrete time t+1 is determined by a randomly chosen, but fixed, Boolean function of the K inputs at time t. The resulting quenched, random dynamics of the network demonstrates two phases: a frozen and a chaotic phase. The authors give an exact solution of the model for connectivity K=1, valid everywhere in the frozen phase and at a critical point, valid for finite as well as for infinite networks. They discuss the network's critical behaviour and finite-size effects. The results for the frozen phase presented complement recent exact results for the chaotic phase obtained for K= infinity .

The anharmonic symmetric oscillator is a frequently used model for physical systems. The recently developed quasilinearisation technique is applied to this model as well as to the simpler one without the quartic term in the potential.

A one-dimensional arbitrary system with quantum Hamiltonian H(q, p) is shown to acquire the 'geometric' phase gamma (C)=(1/2) contour integral _c(P_odq_o-q_odp_o) under adiabatic transport q to q+q+q_o(t) and p to p+p_o(t) along a closed circuit C in the parameter space (q_o(t), p_o(t)). The non-vanishing nature of this phase, despite only one degree of freedom (q), is due ultimately to the underlying non-Abelian Weyl group. A physical realisation in which this Berry phase results in a line spread is briefly discussed.

The behaviour of a three-dimensional kinetic gelation model with regularly placed initiators is determined from Monte Carlo simulations. The authors find marked differences with the behaviour observed when initiators are placed on the lattice randomly. They also look at the effect of introducing repulsion between initiators.

The author presents new low-temperature series results for the 2D planar spin model. High- and low-temperature series results for the energy density are shown to reproduce Monte Carlo simulation results for T<or=0.6 and T>or=1.3 to within one per cent.