Table of contents

The harmonic oscillator is deformed arbitrarily and the properties of the deformed oscillator algebra are studied. The eigenstates and eigenvalues of the deformed oscillator are calculated. This generalized deformed oscillator algebra is used to create nonlinear deformation of the classical SU(2) algebra. The deformed boson realizations of an arbitrary nonlinear SU(2) algebra are calculated. The algebra of the q-deformed oscillator is a special case of the generalized deformed oscillator algebra.

WKB equivalent potentials (WKB-EP) giving the same spectrum as the q-deformed harmonic oscillator with the symmetry SU_q(2) are determined. While in the case of q being real the WKB-EP goes to infinity as one moves away from the origin, in the case of q being a phase the WKB-EP goes to a finite limiting value, thus resembling, for example, the modified Poschl-Teller potential.

Irreducible tensor operators of su(2)q are defined based on the coproduct and are constructed by using the Jordan-Schwinger mapping.

The quantum statistics of a system of free spins in the presence of a constant magnetic field H is interpreted in terms of the representation theory of the simplest quantum group U_q(SU(2)) with q=e^{( mu 0 mu B}kBT)/ being the deformation parameter which has a definite physical meaning. The classical limit q to 1 corresponds to weak magnetic field regime H to 0.

A path integral formulation in the representation of coherent states for the non-compact supergroup Osp(1/2, R) is introduced. An expression for the transition amplitude connecting two Osp(1/2, R) coherent states is constructed, and the corresponding canonical equations of motion derived. A set of generalized Poisson brackets is introduced and interpreted.

The author considers the Euclidean d-dimensional O(N) nonlinear sigma -model in a bounded domain, with prescribed boundary values. A new parametrization of the N-sphere allows one to prove existence and classical differentiability of minimizing solutions under a relaxed smallness condition for the boundary values.

For an electron in the plane subjected to a perpendicular constant magnetic field and a homogeneous random potential the authors investigate the averaged density of states restricted to an arbitrary Landau level They calculate the exact width for an arbitrary random potential. For general Gaussian random potentials they prove the existence of Gaussian tails and determine the decay constants. Furthermore, they derive Jensen-type lower and upper bounds to expectation values of convex functions with respect to the restricted density of states. Using similar bounds they estimate the effect of Landau-level mixing.

The authors have applied the Monte Carlo renormalization group (MCRG) method to study the problem of self-avoiding walks on the Sierpinski gasket family of fractals. Each member of the family is labelled by an integer b, 2<or=b<or= infinity , and when b to infinity both the fractal d_f and spectral d_s dimension approach their Euclidean value 2. They have calculated the critical exponent nu , associated with the mean square end-to-end distance, up to b=80. Their MCRG results deviate at most 0.03% from the available exact results (for 2<or=b<or=9). The obtained data show clearly that nu monotonically decreases with b and crosses the Euclidean value nu =3/4 at b approximately=27, that is, before entering the fractal to Euclidean crossover region that occurs in the limit b to infinity .

For directed compact percolation in two dimensions, the authors conjecture the exact analytical expressions for the percolation probability near a non-conducting wall. The initial 'wet' site is at a varying distance from the wall which is along the oriented 'time' direction. Their results allow an explicit evaluation of the surface-to-bulk crossover in the percolation probability.

A simple multi-affine model for the velocity distribution in fully developed turbulent flows is introduced to capture the essential features of the underlying geometry of the velocity field. The authors show that in this model the various relevant quantities characterizing different aspects of turbulence can be readily calculated. A simultaneous good agreement is found with the available experimental data for the velocity structure functions, the D_q spectra obtained from studies of the velocity derivatives, and the exponent describing the scaling of the spectrum of the kinetic energy fluctuations. Their results are obtained analytically assuming a single free parameter. The fractal dimension of the region where the dominating contribution to dissipation comes from is estimated to be D approximately=2.88.

Ordered phase AB formation and diffusive growth and its boundary evolution on a square lattice are investigated by computer simulation. It is found that the boundary is a self-similar fractal (D=7/4) and its width and length grow as a power law with exponents 1/3 and 1/4, correspondingly. The phase is found to exist over the 0.31-0.69 concentration range. It is shown that a phase formed via a second order-phase transition may have a boundary if the physical properties of the system have a gradient.

The authors propose the procedure of constructing the exact wavefunctions of M magnons on an infinite one-dimensional lattice in the one-parameter model with an exponentially decreasing interaction between spins. For M<or=4 the problem is solved explicitly. The Bethe wavefunctions in the Heisenberg XXX chain appear in the limit of nearest-neighbour interaction.

Analytical and Monte Carlo techniques are used to study a Glauber Ising chain with spatially modulated coupling strengths, under slow cooling approaching absolute zero. The spin configuration freezes if an appropriately-defined effective time does not diverge. The relationship between energy and cooling rate is found to be non-universal with respect to the cooling programme. Rate-matching arguments are used to clarify this non-universality, to categorize freezing behaviours, and to remove ambiguities present in previous discussions.

The remanent magnetization and the remanent energy are calculated exactly for the +or-J spin glass in one dimension with random field and on Cayley trees for sequential dynamics at zero temperature. The method used is an iteration scheme which classifies different kinds of spins. In one dimension a value of ¹¹/₆₀ is found for the remanent magnetization. As expected, the result for the Cayley tree strongly depends on the boundary conditions. The tree with branching number 2 at its 'equilibrium' (i.e. the distribution of the boundary is the same as for the bulk) has a remanent magnetization of ¹/₇ (-33+25 square root 2) approximately=0.336.

The authors present a novel theory of the mixed alkali effect which focuses on the fact that each alkali prefers and maintains its own distinct characteristic environment in the mixed glass. This leads to mismatch between cations and sites recently occupied by unlike cations, and the existence of a memory effect which strongly influences the process of ion migration. This view of glass is consistent with both EXAFS and infrared measurements, and computer simulations reproduce the essential features of the mixed alkali effect including the mobility crossover and the conductivity minimum.

A weight conservation condition for the S-matrix of the braid group representation, from which non-vanishing elements of the S-matrix can be determined, is introduced. A method of weight lattice analysis of representations of Lie algebra is suggested and used to give structures of braid group representations in various cases. This makes it feasible for braid group representations to be obtained by solving the spectral parameter independent Yang-Baxter equation directly, which leads to several new braid group representations.

Some trajectories of the standard group with twist appear to be 'nearly straight lines'. In order to understand the cause of this appearance, a two-dimensional Hamiltonian function with rotational symmetry derived from the map has been studied. For three-, four- or six-fold symmetry, the plane is tiled periodically by sets of parallel straight line energy contours joining the hyperbolic fixed points. For other rotational symmetries, periodic tiling of the plane is not possible, but in some cases there is a strong appearance of quasi-periodic tiling by energy contours corresponding to nearly straight line trajectories of the map. Such energy contours are associated with straight lines along which the variance of the Hamiltonian is a local minimum. For five- and eight-fold symmetry the local minimum value of the variance along such lines decreases quadratically with perpendicular distance from the origin. For seven-fold symmetry, it appears to vary approximately inversely with distance from the origin.

The author considers horizontal Lagrangian sub-bundles of a cotangent bundle T*N that contain the dynamical vector field of a Hamiltonian system. It is shown that if the Hamiltonian is quadratic on the fibres of T*N, then all such sub-bundles can be characterized as solutions to a system of algebro-partial differential equations that are defined solely in terms of vertical quantities. This system of equations has a universal solution. The universal solution is applied to the study of the geometry of the Lorentz force law. It is shown that the dynamical flow of the Lorentz force law on T*N is a geodesic vector field for a metric connection on T*N with torsion. The torsion of this connection is given by an extension of Cartan's first structure equation to non-linear connections.

The authors conclude a research programme developed by various authors in several papers. The basic idea of this programme is to explain the irreversible behaviour of quantum systems as a limiting case (in a sense to be made precise) of usual quantum dynamics. One starts with a system interacting with a reservoir and, in the first attempts to deal with this problem, only limits of observables of the system and deduced master equations were considered. In their approach they study the limits of quantities related to the whole compound system. As a corollary they obtain an explanation of the physical origins of the quantum Brownian motion. They study the low-density limit of a system coupled to a quasi-free boson reservoir in the equilibrium state at inverse temperature beta and fugacity z, through an interaction of the scattering type, i.e. one which preserves the total number of particles of the reservoir.

The authors contrast the classical and quantum dynamics of SU(1,1) coherent states by the use of a positive-definite quasiprobability distribution, a Q-function, defined over these states. For Hamiltonians that are linear in the SU(1,1) generators, therefore coherence preserving, the quantum and classical Liouville equations are identical. This is illustrated for a degenerate parametric amplifier. For an anharmonic oscillator, the quantum Liouville equation contains additional terms containing second-order derivatives with non-positive-definite coefficients. These terms give rise to the quantum recurrences not seen in the classical evolution of this system. It is pointed out that the quantum equation for the Q-function goes over to the classical equation as the Bargmann index, k, becomes large, in agreement with previous semiclassical considerations.

The authors present a variety of theoretical methods to treat the general problem of a quantum particle interacting with a time-dependent barrier. Assuming a sinusoidal form for the time dependence, they discuss the classical, semiclassical and quantum solutions for two specific problems: a barrier whose amplitude is modulated, and one whose mean position changes. While the classical solutions to these problems are quite different, the time-independent quantum solutions appear almost identical. It is only when the particle is modelled as a wavepacket that the quantum solutions behave like their classical counterparts.

The Berry phase for an electron in a one-dimensional box rotated around a magnetic flux line has contributions from the geometry and the magnetic flux, which gives an Aharonov-Bohm effect. For a circular box enclosing the magnetic flux, the Berry phase depends on the boundary conditions.

A differential realization of so(2,1) is shown to be the potential algebra for the one-dimensional systems with the Morse or Gendenshtein potentials. This shows that two classes of Gendenshtein potentials will support the same eigenvalues as the Morse potential, and that the three sets of eigenfunctions may be derived in a common formalism. The potential algebra is then extended to a dynamical potential algebra with operators connecting states both in different potentials and with different energies, giving new dynamical algebras for the Gendenshtein problem. The matrix elements of certain corresponding operators in the three types of system may then be given by a single formula involving so(2,1) Wigner coefficients. The authors also give ladder operators connecting the Gendenshtein potential eigenstates.

The single mode q-Bose field for -1<or=q<1 is shown to be a bounded operator with a continuous spectrum lying in the interval between +or-(2/(1-q))¹2/, thus becoming unbounded in the Bose limit (q to 1). The generalized eigenstates are determined in terms of q-Hermite polynomials whose properties are exploited to obtain orthonormality relations. It is observed that these are connected to important results in combinatorics. It is shown that the eigenstates are products of two q-exponentials of the q-creation operator acting on the Fock vacuum, in contrast to ordinary coherent states which involve only a single q-exponential of the creation operator. A method of representing operators in normal-ordered form based on this representation is described. The spectral resolution is employed to compute the asymptotic temporal behaviour of the particle number expectation value in states produced by coupling the q-Bose field to a source. It is shown that the limit does not commute with the limit q to 1. Specifically it is shown that for O<or=q<1 the asymptotic growth is linear in time in contrast to the quadratic growth for q=1.

The authors consider the self-dual Yang-Mills system along with its hyperbolic version. The author solves the corresponding initial value problem in the second case, and a boundary value problem for the self-dual Yang-Mills equation.

The survival probability P(t)= mod ( phi ₀ mod phi (t)) mod ² for an initial complex Gaussian wavepacket phi ₀ is calculated for the first time by the Lanczos recursion method combined with the QL algorithm without computing the eigenfunctions of the Hamiltonian. Consequently, for an N*N Hamiltonian matrix the correct dynamical behaviour of the system is obtained by carrying out nN² numerical operations rather than N³, where n is the number of Lanczos recursions. The authors show that when phi ₀ is located in the regular region of the classical phase space the ratio n/N is smaller than in the case when it is located in the chaotic regime. This ratio is reduced as the values of t become smaller.

A trapezoidal approximation to the Fermi-Dirac distribution function is used in order to obtain a simple description of semi-degenerate bound-states of fermions. The model coincides with the exact result up to the first correction in the low-temperature expansion while leading to finite equilibrium configurations; thus, it represents an alternative to the use of confining cells or of energy-or density-cutoffs. The atom and the self-gravitating cluster are worked out as explicit examples.

The free energy of condensed matter systems containing fermions coupled to small topological solitons is investigated. The exactly soluble continuum model of polyacetylene, with a hyperbolic tangent soliton profile, is shown to yield non-analytic terms in the free energy of the form lambda ⁿ In lambda where lambda is a parameter proportional to the size of the soliton. A representation of the free energy that is suitable for a small lambda -expansion is derived for general soliton profiles. The non-analytic expansion of the free energy about lambda =0 is found for a general soliton profile for the case of zero temperature.

The authors study the effect of repulsion for self-avoiding walks and random walks from excluded sets. They show, in particular, that the mean displacement away from an excluded infinite needle of self-avoiding random walks in three dimensions has to diverge along the privileged axis as N^sigma , where N is the number of steps and sigma is a sub-leading critical exponent for the two-point function. This exponent has been determined by using a high-precision Monte Carlo simulation ( sigma =0.370+or-0.011). Its knowledge is used to improve the measure of universal quantities, like the exponent nu ( nu =0.5867+or-0.0025, in agreement with the in -expansion estimate and with experimental data) and amplitude ratios. They verify also that for simple random walks the excluded needle introduces instead logarithmic violations to scaling.

A periodic approximant to a two-dimensional (2D) decagonal quasilattice is obtained by the projection method from a 4D periodic lattice, L, which is a commensurate deformation of the 4D decagonal lattice. The Bravais lattice of the approximant is given by the restriction of L onto the physical space, while its space group by the symmetry of the phase vector with respect to the 2D lattice, L_s, which is the projection of L onto the internal space. There exist 12 space groups of the rectangular approximants, pmm, pmg, pgm, pgg, pm1, p1m, pg1, p1g, cmm( Gamma ), cmm(Y), cm1 and c1m, which are derived from the special points or special lines of L_s.

The authors consider a lattice tree model for the collapse of dilute branched polymers in the good solvent regime in which the collapse is driven by a near-neighbour contact fugacity. They describe some rigorous results, including bounds, for the temperature dependence of the reduced limiting free energy and compare these results with numerical estimates derived from exact enumeration data. From the specific heat, they estimate the collapse temperature T_c and the cross-over exponent phi ₀ in two and three dimensions. They find phi ₀=0.60+or-0.03 (d=2) and phi ₀=0.82+or-0.03 (d=3). Finally, they speculate on a possible roughening transition which may occur at a temperature T_r(T_c.

It is shown that a wide class of lattice Hamiltonians in solid-state physics describing the system of N three-dimensional weakly interacting quasiparticles is unitarily equivalent to the one of corresponding Hamiltonians of the system of non-interacting quasiparticles.

The authors consider a kinetic Ising model with ferromagnetic interactions that evolves in time according to two competing Glauber dynamics at different temperatures. The steady states of this non-equilibrium model are studied by using a dynamic pair approximation. When the temperatures are low enough the system orders in a ferromagnetic state, but if one of the temperatures is allowed to be negative the system may have an antiferromagnetic order. They obtain the phase diagram for the case of a square lattice. In this case they calculate the critical exponent v by using a mean field renormalization group method. The numerical results indicate that the model falls into the same universality class of the equilibrium Ising model. They also show that one particular case of the non-equilibrium model studied here is equivalent to the majority vote model.

The authors study the evolution of the concentrations of reactants which perform the irreversible reaction A+B to 0. The reactants are initially separated and are contained in a system undergoing mechanical mixing. They introduce a continuous mixing model which is related to Baker's transformation: the continuous aspect makes it possible to implement mixing in a reaction-diffusion scheme. Furthermore, under very weak requirements, they succeed in presenting analytical expressions for the decay: they show that the initial reaction stages are controlled by mixing so that the concentration of reactants follows the mixing (exponential) time pattern. Furthermore they show that for the usual values of the pertinent parameters (size of the system, diffusion constants, microscopic reaction rates) the standard diffusion-controlled mechanism is recovered only when the reactants are completely mixed.

For an extremely diluted neutral network model designed by the optimal Gardner rule that stores patterns with low level of activity the basin of attraction is studied analytically. The recursion relation for the overlap with a stored pattern is two-dimensional and yields a richer bifurcation scenario than in the case of patterns with symmetric statistics. The bias in the patterns gives rise to ferromagnetic attractors which compete with the patterns.

A neutral network model storing correlated patterns by using a local variant of the projection rule is shown to be equivalent to the Hopfield model.

A calculation of the process of cosmological formation of dyon fermion bound states is reported. It is found that for dyon electron systems the efficiency of bound state formation is about 10% if the dyon charge Q=e/2 and near total if Q>or=e. For dyon-proton bound states it is again near total for all mod Q mod >or=e/2.

Recently the characters of irreducible representations of the Hecke algebra H_n(q) of type A_n-1 were identified with the transition coefficients relating q-generalized power sum symmetric functions to Schur functions (King and Wybourne 1990). Using this result, one obtains an easy combinatorial algorithm for calculating the characters.

The cellular-automaton (CA) AB₂ surface-reaction model of Chopard and Droz (1988) is modified so that the proper stoichiometry in both the B₂ adsorption and the AB reaction is followed. Simulations show that the first-order kinetic phase transition of the Monte Carlo AB₂ model is recovered. A mean-field analysis which predicts a first-order phase transition, is also presented.