Table of contents

Polar decomposition of the generators of the SU(2)_q algebra is performed and its contraction limit to the deformed oscillator algebra is considered. Using the phase operator occurring in the decomposition and its conjugate, the authors write a realisation of the SU(2j+1) algebra into which they embed the deformed SU(2) realisation. Finally they discuss the classical limit of the polar decomposition and the case of the quantum SU(2) dynamics.

It is shown that inverse-power representations of the strongly repulsive potentials sometimes used in molecular eigenvalue and scattering calculations introduce extra errors into the wavefunctions, not normally expected from the finite difference integration formulae used.

It is proved that when the Hilbert space of quantum mechanics is a reproducing kernel Hilbert space of functions defined on the phase space, the generating function of the entropy can be used as a measure of the concentration of the wavefunctions in the phase space. The sharp upper bound obtained for the generating function of the entropy proves that the wavefunctions cannot be arbitrarily peaked in the phase space. The most peaked wavefunctions are those defined by the reproducing kernels.

The authors present the quantum mechanical description of an elliptic billiard which rotates with angular velocity perpendicular to the billiard table. The single particle levels in the rotating frame are evaluated by diagonalisation in the elliptic coordinate basis and the dynamic moment of inertia of the many-particle system is studied as a function of the angular velocity. The irregularities in the behaviour of the moment of inertia are associated with the chaotic dynamics of the classical system.

The authors present a simple model that simulates the dynamics of the long flexible self-associating polymer chains immersed in water and develop a dynamical scaling description for the cluster size distribution function for irreversible aggregation. Each chain may have one or several functional groups at each end that can associate with one another. They find that the chain length is a new scaling variable that appears explicitly in the scaling function. The validity of their scaling predictions is confirmed by extensive computer simulations.

The author describes the critical behaviour of wavefunctions at the Anderson metal-insulator transition in terms of fractal measures. At the mobility edge the wavefunctions have structure on all lengths and they scale with a continuous set of non-related exponents D_q. The results are compared with field-theoretic calculations by Wegner (1980) in d=2+ in .

The mean square distance of a monomer to the origin and the mean square distance of the centre of mass to the origin are studied by Monte Carlo and exact series methods on the square and lattice. The numerical results give strong evidence of a discrepancy with the theoretical prediction of Cardy and Saleur (1989) based on conformal invariance theory.

Induced by phase separation in the lattice gas model a transient percolation phenomenon occurs for concentrations c exceeding a critical curve c^(corr)_p(T, t) which depends both on temperature T and on the time t passed after the quench to a state inside the miscibility gap. A finite-size scaling analysis of extensive Monte Carlo data is carried out to locate c^(corr)_p(T, t) and to show that the critical behaviour of this percolation transition is consistent with random percolation, contrary to previous suggestions.

Using a transfer matrix method the authors derive a series expansion for the percolation probability on the directed honeycomb lattice. The high-density series is obtained to order q¹³. A Pade approximant analysis of the series has been used to estimate the percolation threshold q_c and the critical exponent beta .

Monte Carlo simulations for mixed bond and site percolations on several 2D and 3D lattices show that the critical fractions p^b* and p^s* of bonds and sites follow the relationship (log p^b*/log p_c^b)+(log p^s*/log p^s_c)=1, where p_c^b and p_c^s are the critical fractions in the pure bond and site problems.

The author studies the finite-size effects in the SU(N) and O(2N) Bethe ansatz equations for ground-state configurations. He compares his results with a numerical solution of the associated Bethe ansatz equations.

For a class of one-dimensional nonlinear diffusion equations, where the diffusion coefficient varies as some power of the dependent variable, the invariance group and its Lie algebra are given. The isovector fields which generate the isogroup are then used to derive a 'general' similarity solution. For a particular case the solution can be reduced to a one-parameter group solution which is in full agreement with previously published results.

After reviewing the standard uncertainty relations due to Heisenberg (1927), Robertson (1929), and Schrodinger (1930), as well as the relations of Deutsch, and of Maassen and Uffink-including the so-called entropic relations-the author presents a complete account of the uncertainty relationship between complementary aspects in terms of superspace geometry, an approach not hitherto employed. Two incompatible properties A= Sigma _alpha A_alpha mod alpha )( alpha mod and N= Sigma _nN_n mod n)(n mod belong to a pair of complementary aspects defined by two orthonormal bases ( mod alpha )) and ( mod n)) in the Hilbert space H. If the state is mod psi >, then P( alpha )= mod ( alpha mod psi ) mod ² is the probability of obtaining the value A_alpha in a measurement of A, and P(n)= mod (n mod psi ) mod ² is the probability of obtaining the value N_n in a measurement of N. The two aspects are characterised, relative to mod psi ), by the numbers (so-called purities): pi ₌ Sigma _alpha P( alpha )² and pi ₌ Sigma _n P(n)², both <or=1. He gives a complete characterisation of the uncertainty relationship between A and N (more precisely: between their aspects) in terms of the range of joint values of ( pi , pi ) for arbitrary initial states (pure as well as mixed). A theorem of Lenard is given an alternative proof, employing only elementary (superspace) geometry. The results depend on two angles, phi _m=minimal angle, and phi _M=maximal angle between the two aspects. Exact expressions for phi _m and phi _M are obtained in terms of the overlap matrix Lambda =( Lambda _{alpha n})=( mod ( alpha mod n) mod ²). As a corollary he finds the uncertainty relation for a pure state mod psi ) pi ₊ pi _<or=1+1/g+1-1/g cos phi _m (where g=dim H), and a sharper one for mixed states. pi ₊ pi ₌2 is obtainable if and only if the intersection of the aspects holds a pure state. If phi _m= pi /2 (maximal incompatibility), then pi ₊ pi _<or=1+1/g is a special case of a stronger relation: Sigma _mu ^g_=o pi ^{( mu )}=2, which one obtains for g+1 maximally incompatible aspects by means of a thereom of Ivanovic.

The author applies singular perturbation theory to the study of a damped KdV soliton under the influence of space- and time-dependent external noise. He finds that asymptotically the shape of the averaged solution approaches that of a Gaussian packet, whose amplitude decays and width grows in the same way as in the case of purely time-dependent noise.

The authors study the input impedance of a chain of random resistors chosen according to a 'broad' distribution (i.e. decaying as a power law for large arguments). As a function of frequency, the modulus of the input impedance varies as a power law, while its phase has a well defined value. A simple theoretical analysis, based on estimations of sums of random variables, yields the value of the exponents which are in perfect agreement with numerical simulations, even when logarithmic corrections come into play. The analysis can be extended further to investigate the sample to sample fluctuations of the impedance. Again, agreement with simulations is very good.

Approximate analytical solutions for the dynamical variation of a two-level system with a time-dependent pulse potential are introduced. These approximations are based on the replacement of the potential by a series of step functions and, in principle, can be as accurate as required. The method is extended to many-level transition problems such as those which arise in the theory of surface-ion neutralisation.

The recent classical model of a spinning Dirac particle with Zitterbewegung is generalised to two particles interacting electromagnetically. A variational principle is formulated which leads to a covariant Hamiltonian with separate centre of mass and relative terms much like the quantum two-body Dirac equation. The relative motion has the same form as the spinless case, but with the time-dependent modulated coupling constant representing the spin effects. The canonical quantisation of the theory is immediate.

The general fully supersymmetric approach to the AKNS method is presented. The supersymmetric KdV and MKdV equations are considered as examples. The supersymmetric Backlund transformation is constructed for the supersymmetric KdV equation.

A single degree of freedom, analogous to the coordinate of a particle, is coupled to the density fluctuations of a filled sea of fermions. The fermions live in one spatial dimension and have a linear energy-momentum relation, as may be appropriate in the vicinity of the Fermi surface. By using the equivalence of fermions and bosons in one space dimension, the author shows that the classical limit of the above theory leads to a Langevin equation for the coordinate of the 'particle' in which there are no constraints on the strength of the friction constant. This is in distinction to existing models of fermionic heat baths which have upper limits on the value of the friction constant. The non-classical behaviour of the system is studied by determining the reduced equilibrium density matrix of the particle. The fermions, described by a functional integral over Grassmann fields, are uncoupled from the particle by a change of variables which is equivalent to a chiral gauge transformation. This generates a non-trivial Jacobian which is evaluated in an appendix and leads, in an appropriate limit, to identical results to those found by Caldeira and Leggett (1983) for an oscillator heat bath.

A generalised Sklyanin algebra is studied. The finite-dimensional representations and the centre of this algebra are given with the help of the so-called fusion procedure.

The author examines the possibility of generating squeezed states out of coherent states by external changes of the oscillator frequency. Using the evolution operator method developed by Cheng and Fung, he investigates the evolution of a coherent state of a time-dependent harmonic oscillator as well as its squeezing and coherence property. Two specific models of frequency variation are explored; namely, one whose frequency is exponentially decreasing in time and the other with a periodic frequency. His analyses indicates that the wavefunction of the time-dependent oscillator starts as a coherent state at t=O and evolves as a squeezed state at a later time. It is also shown that squeezing cannot be generated by an adiabatic change of the oscillator frequency because the variances of q and p turn out to be adiabatic invariants, whereas a sudden change can produce squeezing.

The connection between level repulsions and the singularities associated with the analytically continued energy levels is investigated. The authors also conjecture that there are necessarily specific consequences for the state vectors when the statistical analysis of the energy spectrum indicates quantum chaos. A procedure which allows a qualitative assessment of the positions of the exceptional points is suggested. The importance of their distribution for quantum chaos is discussed within this context.

For pt.I see ibid., vol.20, p.6271. The path integral treatment of the hydrogen atom in a hyperbolic space is discussed. The authors show by mapping the radial path integral into the SU(1,1) group manifold that the system has a dynamical SU(1,1) symmetry. The energy spectrum and normalised energy eigenfunctions are calculated. In the flat-space limit, the standard hydrogen spectrum and corresponding normalised wavefunctions are regained.

The exact solution of an Ashkin-Teller model, with layer and spatial anisotropy and Z₂*Z₂ symmetry, is presented. The model is related to the eight-vertex model by a sublattice or orbifold duality and it has the same free energy and magnetisation. The exact solution manifold is a three-dimensional submanifold of the full six-dimensional thermodynamic space. It admits a Z₃ symmetry relating three regimes each exhibiting different partial order. In the spatially isotropic case, these regimes are separated by three lines of continuously varying critical behaviour which coalesce at the four-state Potts critical point. The magnetisations and polarisation are derived using corner transfer matrices giving rise to the appearance of chiral and Virasoro characters. The associated critical exponents confirm the known values. In particular, one obtains the value x_sigma =1/8 for the magnetic scaling dimension and identify the next-to-leading thermal field.

The author uses renormalisation group methods to study the long-distance behaviour of hierarchical (cos phi )₂ lattice field theories in the Kosterlitz-Thouless phase.

Indication is given of the way in which the semiclassical scattering cross section, as a function of the scattering angle, contains hints as to whether the corresponding classical system shows irregular scattering. In the case of classical chaos the frequency spectrum of the interference oscillations of the quantum cross section contains a smeared out image of a classical fractal structure. It becomes better and better resolved in the limit of smaller values of h(cross).

The authors investigate the impedance of two-dimensional electrical networks in the presence of a fractal boundary both by numerical techniques and by renormalisation. Boundaries generated by a Koch-curve construction and by a DLA process are considered. Their results are discussed in connection with measurements of the impedance of electrolytes in contact with a rough electrode.

A general discussion of the kinetics of continuous, irreversible fragmentation processes is presented. For a linear process, where particle breakup is driven by an external force, the authors develop a scaling theory to describe the evolution of the cluster size distribution. They treat the general case where the breakup rate of a cluster of mass x varies as x^lambda . When lambda >0, corresponding to larger clusters more likely to break up, the scaled cluster size distribution, phi (x), decays with the scaled mass, x, as x^-2 exp(-ax^lambda ), as x. For small mass, phi (x) has the log-normal form, exp(-a ln² x), if the breakup kernel has a small-size cutoff, while phi (x) has a power-law tail in the absence of a cutoff. They also show that a conventional scaling picture applies only for the case lambda >0. For lambda >0, they develop an alternative formulation for the cluster size distribution, in which the typical mass scale is determined by the initial condition. In this regime, they also investigate the nature of a 'shattering' transition, where mass is lost to a 'dust' phase of zero-mass particles. They also study the kinetics of a nonlinear, collision-induced fragmentation process. They analyse the asymptotic behaviour of a simple-minded class of models in which a two-particle collision results in either: (1) both particles splitting into two equal pieces, (2) only the larger particle splitting in two, or (3) only the smaller particle splitting. They map out the kinetics of these models by scaling arguments and by analytic and numerical solutions of the rate equations. Scaling is found to hold for different ranges of homogeneity index for the three models.

The lock step model of vicious walkers on a one-dimensional lattice allows each walker at the tick of a clock to move either one lattice site to the left or one lattice site to the right. The only restriction is that no two walkers may arrive at the same lattice site or pass one another. In periodic boundary conditions the partition function and correlation function for this model are calculated exactly. Taking the continuum limit gives an exactly solvable model of vicious walkers undergoing Brownian motion.

The one-dimensional stochastic Lorentz gas is studied in continuous space and in the cellular automata approach. It is shown that the cellular automata method leads to extra oscillations of velocity autocorrelation. The oscillations arise as a result of multiple reflections. The differences between the models become unimportant when the product of the concentration of scatterers and the probability of reflection is small.

The authors study the perimeter length of self-avoiding surfaces on the square lattice. They prove that surfaces with area n, containing h boundary components, have a mean perimeter length of order n. They arrive at this result by studying the incidence of 4-cycles in two-dimensional site animals.

The finite-size scaling spectra of Ising model corner transfer matrices and their generators are studied at criticality. The generators are diagonalised using fermion algebra. The low-lying eigenvalues, given by the zeros of Meixner polynomials, are equally spaced and collapse like 1/log N for large N as predicted by conformal invariance. The asymptotics are obtained using a generalised Euler-Maclaurin summation formula. The shift in the largest eigenvalue is given analytically as pi c/6 log N with central charge c=1/2. The spectrum generating functions, for both fixed and free boundary conditions, are expressed simply in terms of the c=1/2 Virasoro characters chi _Delta (q) with modular parameter q=exp(- pi /log N) and conformal dimensions Delta =0, 1/2, 1/16.

The transport equations in the case of thin insulating films, sandwiched between two blocking electrodes, have been solved numerically. The mobility of the charge is considered to be field and temperature dependent and it is related to the time dependence of the electric field. A trial function for this dependence is introduced. The time evolution of the charge and voltage distribution within the thin film are obtained and the resulting transient current is compared with experimental results.

High-accuracy approximations of the bound-state energies of the potential V(r)=-Ze²/( r+β ) are obtained by means of the dynamical group method. Certain bound-state solutions of the Schrodinger problem are constructed in analytic closed form.

The authors derive an energy function for multibit threshold automata networks (e.g. the Potts model) updated in parallel, and show that their dynamics admit as attractors fixed points and limit cycles of length 2. They compare this result with similar known results on networks of binary automata (e.g. the Ising model).