Table of contents

The objective of this letter is to call attention to a class of special functions which arise from a representation of the SO(3, 2) group that is induced by the representation of its maximal compact subgroup SO(3)(X)SO(2). The author gives the defining differential equations for these functions. A straightforward generalisation of this procedure will be applicable in the SO(p, q) case.

The authors describe a simple and general algorithm to calculate series expansions in enumeration problems to large orders approximately by a Monte Carlo method. It can be used to generate unbiased samples in cluster studies, e.g. linear or branched polymers, random surfaces, etc., in any dimension. They calculate the number of site animals of size n on the square lattice for n <or= 50 and their average size to better than 1% accuracy.

Asymptotic spatial patterns (wavefront solutions) on time-dependent Ginzburg-Landau (TDGL) equations with complex coefficients are discussed. The condition for the existence of a spatial limit cycle solution is found to be given in terms of the propagation velocity of the wavefront solution and the stability criterion for the spatial limit cycle is obtained. The amplitude oscillation of an asymptotic spatial pattern on the TDGL equation (non-linear Schrodinger equation) with purely imaginary coefficients is expressed in terms of Jacobi's elliptic function, while an exact solution of the asymptotic spatial pattern on the TDGL equation (force-free Duffing equation) with purely real coefficients is aperiodic and has a unique propagation velocity.

The action in a one-freedom Hamiltonian system is well known to be invariant under adiabatic change of the Hamiltonian with very small error, provided the frequency of the motion does not vanish. But the frequency can generically vanish: a potential barrier can pass through the energy of the particle. The error incurred in this case has a universal form which is calculated.

The correspondence between a one-dimensional tight binding Hamiltonian and the Lax eigenvalue problem of the classical Toda lattice is applied to a disordered chain. A model with specific diagonal and offdiagonal disorder is solved by using canonical spectral variables. An exact analytical expression for the density of states is presented.

A regular-random fractal, intermediate between statistical fractals and deterministic fractals, is presented to imitate the geometric texture at and near percolation threshold. The model is constructed on the square lattice via a rule of bond occupation with use of position space renormalisation group. It shows the typical percolation behaviour as a function of a parameter p (the bond concentration). The critical bond concentration, correlation length exponent and scaling property of the cluster size distribution are found. The scaling relations between critical exponents for cluster numbers and structure are shown to be exactly satisfied. The fractal dimension of its cutting bonds agrees with the inverse of the connectedness length exponent at criticality.

Considers the critical behaviour of a particular set of onedimensional self-dual models with Z(N) symmetry (N <or= 5). The critical indices are evaluated using standard finite-size scaling and by exploiting their relations with the mass gap amplitudes predicted by conformal invariance. The results strongly suggest that the recently introduced Z(N) quantum field theory is the underlying field theory for these statistical mechanics models.

The corrections to the finite-size scaling behaviour of the eigenvalues of the transfer matrix of a critical theory defined on an infinitely long strip of finite width, which occur when the Hamiltonian contains a marginal operator, are computed using conformal invariance. They show a calculable universal logarithmic character. For the four-state Potts model they agree with numerical data.

The authors show that the anomalous scaling behaviour of the moments of the wavefunction at the threshold of Anderson localisation implies a log-normal distribution for the probability mod psi mod ² at first order in epsilon =d-2. They discuss this result and its implications critically and are led to conclude that the upper critical dimension of Anderson localisation is infinity.

Using a Monte Carlo simulation, the authors have studied the kinetics of the domain growth in the dilute random-field Ising model with Glauber dynamics in two dimensions, following a quench from a very high temperature to a low-temperature unstable state. The data strongly suggest a breakdown of self-similar dynamical scaling.

A simple model (based on conformal mapping) is constructed to simulate the two-dimensional growth of needle crystals growing out regularly from a common point. The resulting patterns are very similar to those which were obtained by other methods. The radius of gyration exponent beta is measured and found to be in the range 0.58-0.65, independent of the number of needles (n). It is shown that for large patterns, beta asymptotically approaches a value of 2/3. To compare the patterns with the DLA aggregates, the average slope ( alpha ) was measured for the density-density correlation function plotted logarithmically (log r to log C(r)). The relation between alpha and beta , which is characteristic of DLA fractal aggregates, is satisfied only in the case n = 8.

A Lie algebraic approach to quadratic parametric processes in quantum optics and quantum acoustics is presented. In this approach the Heisenberg-Weyl and symplectic dynamical algebras are used to obtain a general exact solution for the time evolution operator. The solution is then applied to describe quantum mechanically the parametric process of backward-wave echo generation in dielectrics.

The complementarity relation between the U(p, q) and U(n) Lie groups enables the authors to reformulate the SO(6, 2) model of SU(3) in terms of a U(1, 1) group and to outline its generalisation to a family of n -2 models of SU(n), for n >or= 3, respectively associated with U(n-2, 1), U(n-1, 2),..., and U(1,n-2) groups.

The largest symmetry group of the Schrodinger equation, the so-called Schrodinger group, is analysed in connection with invariant 2forms, 1-forms and (0,2)-symmetric tensors through the Beckers-Harnad-Perroud-Winternitz global method (1978) developed in the relativistic conformal context. Invariant fields and potentials are obtained and discussed in the physical context of Schrodinger electromagnetism through the corresponding infinitesimal method. Specific attention is paid to magnetic monopole dynamical symmetries. These results are obtained in correspondence with the subalgebra classification determined by using the Patera-Winternitz-Zassenhaus algorithm (1975). The accent is put on the maximal subalgebras of the Schrodinger algebra.

Using the outer product reduction coefficient of the ordinary permutation group, the authors have calculated the Clebsch-Gordan coefficient for the non-special Gel'fand basis of the graded unitary group SU (m/n) for the five-particle system.

A number of new infinite series of S functions are described in terms of their generating functions and S function content. Applications to the character theory of non-compact Lie groups are noted.

The Schrodinger equations of rotating harmonic, three-dimensional and doubly anharmonic oscillators and a class of confinement potentials are examined simultaneously with the biconfluent Heun equation. The eigenvalue problems lead the authors to compute a transcendent constant function of the corresponding physical parameters.

Recently, Nandakumaran (1985) has proved that under the discrete-time quadratic map x/sub /t₊₁=4x/sub /t(1-x/sub /t), all the probability density distributions r₀⁽n⁾(x)=(x/sup /n(1-x)/sup /n)/(B(n+1,n+1)) 0<or=x<or=1 n=0,1,2,... converge towards an invariant limit density associated with the map when t tends to infinity. The purpose of the present paper is to generalise this result. Starting from the Fourier series expansion of any real single-valued function which satisfies the Dirichlet conditions in (0, pi /2), a broad class of normalised initial functions (w₀(x) mod 0<or=x<or=1) is obtained, each of which is the sum of a convergent series involving the Chebyshev polynomials of both kinds. The evolution equation for this class of functions under the map mentioned is found explicitly. It is shown that the absolute convergence of the series expansion of the symmetric part in an initial w₀ function is a sufficient condition to ensure an evolution for t to + infinity towards the same invariant limit density as obtained by Nandakumaran with the infinite set of particular initial densities r₀⁽n⁾(x). Some auxiliary results related to the treatment of the considered problem are also presented.

A numerical solution is obtained for the Laplace-transformed backward Kramers equation, from which the mean first-passage time may be obtained. The main difficulties are associated with (a) the parabolic nature of the time-development operator and (b) the existence of a double structure in the solution near the absorbing barrier. Both of these difficulties are resolved by computational methods derived from boundary layer theory. The reliability of the method is assessed by comparing its results with an earlier analytic solution for the case of a uniform force field. The authors also present the results for a harmonic force field, for which no analytic solution is yet known.

Ion acoustic solitons affected by the drift motion of the electrons along the direction of the magnetic field have been investigated. The ranges of the existence of solitons of different velocities and the limiting value of drift velocity are reported.

The classical limit n>>0 is investigated for a two-level system coupled to a boson mode. The usual formulations are found to neglect a shift lambda ² of the spectrum where lambda is the strength of the coupling. A scaling law for the highly excited energy levels is derived and tested numerically. An insight is obtained into the characteristic properties of the energy level distribution which is of interest in connection with possible 'quantum chaotic' behaviour. The interpretation of the asymptotically effective Hamiltonian as a one-dimensional tight-binding model for the two-band Stark ladder is discussed.

A new class of solutions of the time-dependent Schrodinger equation is found for the two-state problem often encountered in quantum optics, magnetic resonance and atomic collisions. The authors use the Riemann-Papperitz differential equation to find exact solutions in terms of hypergeometric functions. They consider only cases in which the final occupation probabilities are elementary functions of the parameters of the model.

The electronic structure of the many-electron atom in spherical space is investigated. A new expression of the bi-electronic repulsive integral is given. The ground state energy levels of the helium and lithium atoms are obtained. A parametric duality between the radius of curvature R and the atomic number Z is given prominence in the calculation of the screening effect.

The authors present a quantum electrodynamic model, soluble in the dipole and rotating wave approximation, for a three-level atom interacting with a two-mode resonant radiation field through the multiphoton transition mechanism. Population dynamics and photon statistics in this Jaynes-Cummings type model are examined.

If the free electron gas is enclosed in a box of finite volume the Dirichlet boundary condition imposed on the wavefunction modifies the density of states (it increases the energy levels). This is also true in presence of a uniform magnetic field and gives rise to the perimeter corrections chi ' to the Landau diamagnetic susceptibility chi ₀. The author analyses the effect for the zero-field susceptibility by a Green function approach rather than by enumerating the energy levels. The perimeter contribution chi ' to the susceptibility is always positive (paramagnetic). The relative correction chi '/ chi ₀ is given by (apart from a constant of order unity) l*surface area/volume, where l is a characteristic length and is equal to the thermal de Broglie wavelength at high temperatures and to the Fermi wavelength in another extreme of complete degeneracy. Thus the effect may be observable in small metallic particles of size 10-100 AA, in particular if the electron effective mass is small such as, e.g., in bismuth.

Low concentration series are generated for moments of the percolation cluster size distribution, Gamma _j=(s^j-1) (s is the number of sites on a cluster) for j=2, . . ., 8 and general dimensionality d. These diverge at p_c as Gamma j approximately Aj(p_c-p)^{- gamma j} with gamma j= gamma j= gamma +(j-2) Delta , where delta = gamma + beta is the gap exponent. The series yield new accurate values for Delta and beta , Delta =2.23+or-0.05, 2.10+or-0.04, 2.03+or-0.05 and beta =0.44+or-0.15, 0.66+or-0.09, 0.83+or-0.08 at d=3, 4, 5. In addition, ratios of the form A_jA_k/A_mA_n, with j+k=m+n, are shown to be universal. New values for some of these ratios are evaluated from the series, from the epsilon expansion ( epsilon =6-d) and exactly (in d=1 and on the Bethe lattice). The results are in excellent agreement with each other for all dimensions. Results for different lattices at d=2, 3 agree very well. These amplitude ratios are much better behaved than other ratios considered in the past, and should thus be more useful in characterising percolating systems.

Algebraic criteria which allow the vertices of two- and three-dimensional Penrose tiling patterns to be specified are presented. The application of these criteria to the interpretation of electron microscope observations from quasicrystalline structures is discussed.

A problem of the configurational properties of a long flexible polymer chain with a quenched disorder is considered. The chain is assumed to be randomly constructed from monomers of two different kinds with different constants for the two-body interaction. Near the theta point, i.e. when the average interaction of monomers is small, the spatial correlation of the repulsive and attractive monomers of different kinds leads to an increase of effects of the disorder on large scales. There is also the competing effect of the repulsive three-body interaction which tends to screen the effects of disorder on large scales. For both effects the upper critical dimension is d_c=3. A solution of the renormalisation group equation indicates that there always exists a critical scale at which the relative dispersion of sizes of polymers with different random sequences of monomers becomes of the order of unity. The magnitude of this critical scale depends strongly on the relation between the constant which characterises the dispersion of the two-body interaction B₀ and the constant of the repulsive three-body interaction V₀. If B₀<³/₃₂V₀ at each physically attainable scale the effects of screening are prevalent and the dispersion of sizes of polymers with different sequences of monomers is small near the theta point. If the reverse inequality holds, the dispersion of sizes becomes of the order of unity near the theta point.

A renormalisation group approach is developed for Delaunay percolating systems in two and three dimensions using a scaling transformation for a finite lattice in real space. Considering various renormalisation transformations for two- and three-dimensional Delaunay lattices, the authors determine the behaviour of the probabilities under a scale transformation and calculate the fixed point and connectedness length exponent. The fixed points for the two-dimensional bond lattice and the three-dimensional site lattice are 0.3229 and 0.1443 respectively, which are in excellent agreement with results of Monte Carlo simulations. The fixed point for the two-dimensional site lattice gives the value ¹/₂ for the critical percolation probability which is equal to the known result for a fully triangulated lattice.

Two models for random two-conductor mixtures by diffusion processes are considered in the one-dimensional case. It is shown that the two models are not in the same universality class. This is seen to be an artefact of systems with p_c=1, however. A scaling theory proposed earlier is tested for this simple one-dimensional case and confirmed.

In a recent paper Ma and Xu (1984) have obtained a set of equations for the embedding of an SO(4) pseudoparticle with vanishing energy-momentum tensor in SU(N) Yang-Mills theory and have obtained one particular solution. The authors completely solve these equations here.

It is demonstrated that the contraction of Lie algebras can be viewed as the procedure that underlies limiting distributions in probability theory. The consequences of such an interpretation are discussed.

It is shown how integrals of certain combinations of solutions to a second-order differential equation may be found in an elementary way. In particular, an integral involving Airy functions which has occurred in recent studies of the asymptotic behaviour of the gap in the Mathieu equation is evaluated.

A traditional Monte Carlo simulation of a 1000*1000*1000 simple cubic lattice gave a bond percolation threshold near 0.2494 and a site percolation threshold near 0.3116 with probable error bars near 10^-4.

Using the form prescribed by Moreno and Zepeda for the N-dimensional Yukawa potential and employing the method of 1/N expansion we have obtained in a recent paper the energies of the ground state and the first excited state of a three-dimensional Yukawa potential. In the present addendum we introduce a modified prescription for the N-dimensional Yukawa potential which enables us also to calculate the energies of the higher angular momentum states. Our results are in very good agreement with the accurate numerical values.