Table of contents

The authors consider a lattice system described by a set of nonlinear differential equations, and discuss how general lattice properties and/or symmetries can allow the reduction of this problem to a lower-dimensional one, and possibly grant the existence of periodic or wave-like solutions.

Dirac's free particle equation originated in an attempt to express linearly the relativistic quadratic relation between energy and momentum. The authors introduce a Dirac equation which, besides the momentum, is also linear in the coordinates. They call it the Dirac oscillator because in the nonrelativistic limit it becomes a harmonic oscillator with a very strong spin-orbit coupling term. The eigenstates and eigenvalues of the Dirac oscillator can be obtained in an elementary fashion, with the degeneracy of the latter being quite different from that of the ordinary oscillator. They briefly mention the symmetry Lie algebra responsible for this degeneracy and the generalisation of the problem to many-particle systems.

The covariance properties and the Foldy-Wouthuysen and Cini-Touschek transformations for the recently proposed Dirac oscillator are found. The charge conjugation, parity and time reversal properties are established. A physical picture as an interaction of an anomalous (chromo) magnetic dipole with a specific (chromo) electric field emerges for this system. This interaction suggests an alternative confinement potential for heavy quarks in QCD.

The application of functional formalism to the description of turbulent diffusion is shown. The problem of the turbulent diffusion is formulated on the basis of a variational principle for the Phi equation in the appropriate functional space. The existence of an exact constant of motion, as a consequence of this principle, is shown.

Build-up of roughening fluctuations in an interface at the wetting (unbinding) transitions can be measured by the free energy of a step-like formation due to a stepped substrate. For the 2D solid-on-solid model of line interface unbinding, exact calculations are presented. It is found that the step free energy vanishes linearly as the wetting transition is approached, as opposed to the well known quadratic vanishing of the interfacial binding energy (per unit length). Step-step interaction free energy is also calculated, for large step separation.

The authors extend the definition of droplets in Ising and Potts models to the case of an external field different from zero. They also find exact relations between thermal properties and connectivity properties which show why, in mean field, the mean cluster size does not diverge as the susceptibility when the critical temperature is approached from below.

For pt.I see Discrete Appl. Maths., vol.19, p.367 (1988). The author reports the results of taking the reciprocal of generating function for self-avoiding walks on the plane square lattice with end-to-end distance specified. In a previous paper it was shown that this produces the generating function for irreducible two-point Mayer clusters. This function is found to be far simpler than the original generating function. It is found that a renormalisation or self-consistent field calculation should give very good results and that further small corrections due to the existence of 'traps', can also be computed. It is further found that the generating function has an unphysical singularity very near to the physical singularity and that this accounts for difficulties of series analysis that have been experienced, for example, by Guttmann (1987).

The authors have looked for the signature of chaos in the Abelian Higgs model and SO(3) Georgi-Glashow model, which possess vortices and monopole solutions respectively. On applying Painleve analysis they find that most of the type-I region of superconductivity in the Abelian Higgs model and lambda >2g² region in the Georgi-Glashow model is nonintegrable (here lambda is the Higgs coupling while g is the gauge coupling constant). Further using the Toda-Brumer criterion they find that the critical energy for the onset of chaos is E_c=(11/108)(c₂²/c₄) and E_c=m⁴/54 lambda in the Abelian Higgs model and Georgi-Glashow model respectively.

The lattice gas model is studied as a function of dimensionality by a combination of analytical and numerical techniques. It is shown that for d<or=2 the order-disorder transition density rho _c is higher than the maximal filling density for a random filling process rho _r, while for d>or=3, rho _r> rho _c. The existence of a disordered metastable phase with rho > rho _c is discussed in view of these results.

For pt.I see ibid., vol.28, p.2323 (1987). Previously the author considered a problem arising in the theory of quasicrystals pointed out by Trebin. It amounts to the computation of a basis of a subspace of (X)^2dR³ d in N characterised by certain symmetries. The solution was given for d<or=3. Now the author is able to describe such invariants for arbitrary d.

A stochastic model for a dynamical system containing a great number of randomly interacting species is introduced. In the limit of infinitely many interacting species this model becomes exactly solvable and shows a stability-instability transition driven by the typical interaction strength and the noise parameter. The dynamical behaviour of the system in the stability region of the phase diagram is investigated and compared with numerical simulations. The implications of the results on natural ecosystems are discussed.

The nonintegrability of three Hamiltonian systems obtained by simplifications of the Yang-Mills classical field theory is proved by application of two theorems of Yoshida (1987, 1988). It is also shown that for the two four-degree of freedom systems there exists a set of three independent integrals of motion in involution.

The stationary electromagnetic field produced by a rapidly rotating, magnetised and conducting sphere is discussed. The field is determined by solving Maxwell's equations for an accelerating medium and reduces in the nonrelativistic limit to the expected unipolar induction field of a slowly rotating spherical magnet.

The empirical observation that the single-particle energies generated by solving the Schrodinger equation for a deep bell-shaped potential form almost perfect rotational bands is examined within a semiclassical approximation. Formulae are deduced to reconstruct the potential V(r) which gives rise to a bound-state energy spectrum of the form E( nu , lambda )=f( alpha nu + lambda )+ beta lambda ² where alpha and beta are arbitrary constants, nu -¹/₂=n is the number of internal nodes in the radial wavefunction and lambda -¹/₂=l is the angular momentum quantum number. The authors further find that the unspecified function f is, in fact, determined by alpha and beta . A discussion is given of the conditions governing the possibility of making the inversion to obtain V(r).

Berry (1984) discovered that an eigenstate undergoing an adiabatic evolution in the parameter space will acquire a topological phase. Aharonov and Anandan (1987), on the other hand, showed that an eigenstate transporting round a closed circuit in the projective Hilbert space suffices to generate the topological phase. The authors employ the evolution operator method to study the propagation of an eigenstate. They show that Berry's phase can be represented as a closed path integral in the Hilbert space and is independent of the choice for the base ( mod n(t))). The treatments of Berry, and Aharonov and Anandan, are shown to correspond to two different choices of the base. Therefore their two approaches are unified; they have acquired a more general viewpoint on the origin of Berry's phase.

The eigenvalue spectrum E_l(V,n,α) of normalisable states of the equation (αℏ/im)ⁿϕ^(n-1)(x)-ℏ²/2m) ϕ"(x)+(V(x)-E)ϕ(x) = 0, recently proposed as a quantisation of classical forces -αxⁿ - δV/δx, is discussed for even n. For n = 4 and a rectangular potential well an analytic argument shows that bound states for 0 < α < , → 0, can exist only if E_l(V,4,α) → E_l⁽⁰⁾(V) as α → 0 where E_l⁽⁰⁾(V) are the eigenvalues of the corresponding conservative Schrodinger equation (α = 0). Numerical results indicate E_l(V,4,α) - E_l⁽⁰⁾(V) = c_l(V) α² + O(α³) as α → 0. For n = 2, E_l(V,2,α) - E_l⁽⁰⁾(V)= 1/2 α²ℏ²m^-3 is exact and independent of V(x), and at a potential step (to the left since α > 0) a continuous band of normalisable states exists instead of the scattering states for α =0.

The authors present a thorough analysis of the removability of the Berry phase. They show that for any system they can, by properly restricting the parameter space turn the geometrical phase into an extra shift of the dynamical one. However, this extra shift in the dynamical phase retains its truly geometrical character. The removability is thus only apparent and the geometrical phase can still be measured in interference experiments, as indeed has already been done several times.

Utilising the intimate connection of a charged particle in a homogeneous magnetic field to that of a harmonic oscillator, the authors demonstrate explicitly that the eigenvalue spectrum for the matrix operator S. pi for arbitrary spin in a uniform magnetic field in the z direction is governed, for its discrete part, by that of a Hermitian matrix defined on the space of the particle number and spin operators and is thus constrained to be real for any intensity of the external magnetic field. The present analysis generalises to arbitrary spin their previously reported results applied to the case of a spin-1 particle.

An example of billiards in a moving boundary, which exhibits regular and chaotic motion, has been discovered recently. The specific system is the motion of a particle inside a circular billiard table which rotates uniformly about a point on its edge. At one energy the motion becomes ergodic, but is regular for very low and for very high energies. The author studies the quantum mechanics of the system, and calculates numerically the energy spectrum. In the semiclassical limit, he shows that for energies near to where the system is ergodic, the level spacing distribution is Wigner. As h(cross) to infinity the system becomes equivalent to the stationary case, but the spectral statistics for low energies is indistinguishable between Wigner and Poissonian. For some intermediate h(cross) he finds that the distribution becomes very Poissonian even though the energy levels used are in the chaotic region of the classical motion.

A new technique is proposed which allows one to avoid the space- and UV-cutoffs when quantising field-theoretic models integrable by means of the quantum spectral transform (R-matrix) method. The technique is based on a new definition of the monodromy matrix for the infinite interval. For the nonlinear Schrodinger equation (repulsive case, zero density) the known excitation spectrum is reproduced.

The quantum treatment of an intrinsically chaotic model scattering system originally studied by Gutzwiller (1983) is extended to include the time delay and to make explicit the zeros of the Riemann zeta function in the mathematical expressions for the scattering matrix S and the time delay. The system consists of a particle moving on a two-dimensional surface of constant negative curvature. The scattering in this unusual system is dominated by resonances associated with poles of S in the complex momentum plane, the real parts of these poles are one-half of the imaginary parts of the Riemann zeros. The resonances have a constant width but their average spacing varies with the momentum. The focal point is the manifestation of chaotic behaviour in the time delay. Features considered in this regard include: the momentum dependence of the time delay in regions of overlapping and isolated resonances, the decomposition of the time delay into an average (dynamical) component and a fluctuating (chaotic) component, and characterisation of the fluctuating component by its autocorrelation function.

The authors study the quantum mechanics of two-dimensional spaces of constant negative curvature. The connection between different formulations is analysed. They derive the propagator and find that in the semiclassical approximation, the time evolution of observables exhibits similar features as in the classical case.

The term by term separability of the Sturmian function representation of the Coulomb Green function is exploited to construct expressions for the single and double transforms of the corresponding outgoing wave Green function by the form factors of the Graz separable potential.

Single- and double-slit diffraction patterns are obtained by means of path integrals. The slit function is taken to be unity. This permits the study of the single-slit case as a limit of the double-slit case when the slit width is equal to the slit separation. Introduction of a whisker of flux at the midpoint of the space between slits gives the Aharonov-Bohm shift. In the case of a wide single slit, one obtains conditions approaching those postulated in calculations similar to the scattering calculation of Aharanov and Bohm (1959).

The authors investigate the winding angle and mean square end-to-end distance of lattice trails on a square lattice, by both exact enumeration and Monte Carlo methods. They find similar behaviour to that of self-avoiding walks, to the extent that the complete distribution is found to agree with the recent exact calculation of the winding-angle distribution function for self-avoiding walks by Duplantier and Saleur (1988), who found a Gaussian distribution. The exponent nu was found to be indistinguishable from 3/4.

Two properties of self-avoiding walk networks, constructed by connecting the nearest-neighbour points of self-avoiding walks on the square lattice, are studied numerically. The average end-to-end linear part of such networks is investigated, and is found to be proportional to the length of the walk. Secondly, random walks on this structure are studied and the associated spectral dimension of such networks is estimated to be 1.03+or-0.03, which implies that the effect of such connections affects the spectral dimension only weakly, if at all.

The authors calculate the scaling behaviour of the magneto-resistance of a regular three-dimensional (3D) fractal network model in a weak magnetic field. The method used is a discretised version of a previously described continuum theory. Exponents are calculated that characterise the power-law dependence of both the transverse and the longitudinal magneto-resistance on the total size. Results are compared with a previously published prediction of scaling theory for the magneto-resistance near the percolation threshold.

The author considers an infinite sequence or one-parameter family of 1D maps (f_m(x)) which converges uniformly to a target map f(X). The parameter m can be continuous or discrete. f_m(x) have non-linear dependence on m. The target map f(x) generates fully chaotic dynamics. The author analyses the dynamics generated by the sequence (f_m) as the target map f(x) is approached through this sequence. The first results of computer experiments regarding two special cases of the above situation are presented.

The W sequence, an ordering of maximal itineraries for circle maps, is constructed by means of the Farey transformation and the gluing operation. The similarity in the dynamical behaviour of the maps is explicitly described in terms of itineraries. A class of badly ordered itineraries, termed misbehaved, is defined with the language of symbolic sequences. A misbehaved itinerary is shown to imply a rotation interval.

Cluster expansion methods are applied to calculate 'high-temperature' series for the vacuum energy, the susceptibility and the mass gap for the Ising model and the O(2) and O(3) Heisenberg models in (1+1) dimensions and (2+1) dimensions. Critical points and critical indices are estimated for the line, the square and the triangular lattices. The results demonstrate universality with the normal Euclidean versions of these models, within errors.

The study of the universal finite-size scaling amplitudes of interfacial free energies in Monte Carlo simulations provides an accurate method of determining the critical temperature and the universality class of phase transitions. These universal amplitudes can replace the critical exponents to identify universality classes. The authors use Bennett's optimisation scheme (1976) for the calculation of the interfacial free energies in Monte Carlo simulations. Tests for the Ising and three-state Potts model confirm the effectiveness of this method.

The author presents results of Monte Carlo simulations for directed percolation in 2+1 dimensions very close to the percolation threshold. His values for p_c for bond and site percolation on the BCC and simple cubic lattices are more precise than previous estimates by one to two orders of magnitude. The improvement on the critical exponents describing the behaviour at p_c is less impressive, but they are also substantially more precise than previous ones.

The coherent medium approximation is applied to study the frequency-dependent diffusion constant in two- and three-dimensional ant-termite like mixtures, where three types of jump rates, 'normal' (metallic), 'superconducting' and 'insulating' are distributed randomly. The DC diffusion constant vanishes when (p+r)<or=p_c and is proportional to (p+r-p_c)/(r_c-r) when p+r>p_c, where p and r are the probabilities that a given bond is metallic and superconducting, respectively, and z is the coordination number of the lattice, p_c=r_c identical to 2/z. In the low-frequency limit, the real and imaginary parts of the AC diffusion constant go as A(z,d,p,r)f( omega ) and B(z,d,p,r)g( omega ), respectively, a behaviour similar to the pure ant limit. The behaviour of f( omega ) and g( omega ) is examined below, at and above the percolation thresholds, in both two and three dimensions. The coefficients A and B of these leading terms diverge as r to r_c (superconducting region percolates), indicating enhancement in the dielectric constant.

The authors consider the dynamics of invasion percolation for different models. This process exhibits scaling behaviours which, in the simplest case, can be related to some critical static properties of usual percolation. In particular, the concept of bursts is shown to be of fundamental importance. The critical exponents they derive are checked numerically on a case of invasion percolation where growth is restricted to the hull of the existing cluster.

A large class of potentials can be solved algebraically by using supersymmetry and shape invariance. The authors apply operator transformations (f transformations) to these algebraically solvable problems to obtain a larger class of solvable potentials-the Natanzon class of potentials which are not shape invariant. The important condition (which is independent of supersymmetry) for finding new potentials from old ones using operator transformations is that the resulting Schrodinger equation has a potential which does not depend on the state. As a special case of the f transformation they rederive the previously known connection between the 3D harmonic oscillator, the hydrogen atom and the Morse potential. They also discuss the lack of commutivity of SUSY and the f transformations.

A gauge transformation of the minimal-coupling Hamiltonian for a non-relativistic hydrogen atom is induced by a generating function that rescales the charges and their positions. The result is consistent with Maxwell's equations and the Lorentz force formula, and contains the Power-Zienau-Woolley Hamiltonian as a special case. The dipole approximation simplifies the Hamiltonian, given a mixing of p.A and q.E interactions.

The sl(4,R) Lie algebraic treatment of the Winger SU(2) matrices (1959) is extended by analytical continuation to the Bargmann SU(1,1) matrices (1947) corresponding to the positive discrete series irreducible representations. It is then used to obtain an sl(4,R) dynamical potential algebra for the negative-energy solutions of the second Poschl-Teller equations (1933).

As an alternate way of measuring the lacunarity of Sierpinski carpets, the authors have computed the site to site fluctuation of the quantity M(n), namely the total number of lattice sites within the chemical distance of n. In long scales of n, the resultant lacunarity agrees qualitatively with the results obtained in the past using the covering method. In short scales of n, however, the lacunarity shows a different trend among fractals; it varies monotonically with the Ising correlation length exponent, a trend which was sought before but not found with the previous method. This applies only for the carpets of the first kind.