Table of contents

A new tool for deriving Regge trajectories and energy eigenvalues for ground states of central potentials is applied to the Klein-Gordon equation. Based upon the h(cross)-expansion, the simple recursion formulae are presented. The problems of the pi -mesonic atom and funnel-shaped potential are treated with this technique.

Two classes of analytic refractive-index profile P²(z, in ), whose reflection coefficients r are zero for all values of a parameter in , are studied as in to 0. The aim is to understand why r=0 rather than r varies as exp(-1/ in ) as for generic profiles. The authors find that reflectionlessness is a consequence of the fact that transition points of P² (zeros or poles in the complex z plane) form tight clusters (whose size vanishes with in ) which can be regarded neither as coalesced nor well separated. Expansion near a cluster yields the local wave not as the usual Airy function, whose Stokes phenomenon generates reflection, but as Bessel functions of half-integer order (fake Airy functions) which are exactly trigonometric functions with no Stokes phenomenon and so no reflection.

The author determines the role of domain walls travelling along the backbone of an RNA double stranded helix in proton exchange events. The results hold for (polyU.polyA) duplexes (U and A are bases paired by means of Watson-Crick H-bond interactions). The author proves that the density of domain walls is increased by the same phonon excitations which would enhance proton exchange with the solvent. The identity of the catalytic groups in the process is established.

The authors announce results which prove the existence of the spectral dimension d_s for Sierpinski carpets in two dimensions. Their method employs the Einstein relation zeta =d Omega -d_f. Using this, numerical calculations of the resistance of approximations to the Sierpinski carpet yield an accurate estimate for d_s.

The authors introduce a new approach to the analysis of growing unstable interfaces in order to characterise the degree of their randomness. Highly ramified viscous fingering patterns are digitised and the related surface curvature data are evaluated using techniques common in the investigations of dynamical systems. Their results for the associated correlation dimension indicate that the experimentally observed deterministic patterns correspond to a spatial behaviour which is more complex than a low-dimensional chaotic geometry.

The authors show that in the three-dimensional Ising model at the critical temperature the fractal dimension of the touching damage, determined by the box counting method, asymptotically converges to the expected value d_f=d- beta /v. In contrast, the finite-size scaling analysis indicates for the touching damage an effective fractal dimension whose value is 20% smaller. The expected d_f is, however, recovered again when instead of the system size L the quantity L/ln L is used as a scaling variable. This behaviour could be explained by the recently discovered dynamical multiscaling.

The author considers the linear divergence which appears in field-theoretical computations of the 2D Ising spin-spin correlation function. By expressing the correlator in terms of fermionic determinant, the author shows that this divergence is connected with a certain gauge transformation of the fermion model. The author proves that if gauge invariance is preserved then the divergent term is not present in the final result.

The dynamics of the n-component Ginzburg-Landau model with non-conserved order parameter (model A) are considered following a quench to zero temperature. The correlation function of the order parameter field is found, in the 1/n expansion, to have the asymptotic scaling form C_k(t,t')=t'^d2/(t/t')^lambda 2/f(k²t,k²t') for t>>t', with f(0,0)=constant. The new exponent lambda is calculated to O(1/n) for general space dimension d, and has a non-trivial dependence on n and d.

The storage capacity of a Hopfield model with patterns forming a two-level hierarchy is calculated without reference to a special learning rule. If the number of different classes becomes very large strong correlations within the classes decrease the storage capacity in striking contrast to what is known about one-level hierarchies, i.e. patterns with magnetisation.

The authors extend the Lax pair for Kowalewski's top (KT) obtained by Haine and Horozov (1987) to the Lie algebras o(4) and o(3,1). The results are expressed in terms of a solution of the Neumann system. They derive formulae for the action variables by analogy with the e(3) KT case and obtain a useful representation for the equations of motion that demonstrate explicitly the separation of variables.

Positive discrete series irreducible representations of the semidirect sum Lie algebras wsp(2N,R)=w(N)+sp(2N,R) are given in wsp(2N,R) contains/implies sp(2N,R) contains/implies u(N) bases. The branching rule for the reduction of a wsp (2N,R) irreducible representation into a direct sum of multiplicity-free sp(2N,R) irreducible representations is accounted for by the explicit construction of a wsp(2N,R) contains/implies sp(2N,R) contains/implies u(N) basis of vector-valued holomorphic functions.

In a previous paper the author introduced a path integral technique based on birth and death processes. The method provides an analogue of the Feynman-Kac formula which is associated to the Hamiltonian semigroups of some boson systems. The author briefly demonstrates this technique in a general form. Furthermore, the author studies a simple spin-boson model for which the author obtains some estimates on ground-state energy and spin-flip rate.

In the quasiclassical limit, the MKP hierarchy and its Lax representation turn into a modified Benney hierarchy and its Poisson representation. A Miura map is constructed, and shown to be canonical, from the modified Benney hierarchy into the unmodified one. The modified hierarchy is given both hydrodynamical and kinetic representations, and the Miura map is given a kinetic form. Explicit combinatorial formulae are proved for the infinite number of conserved densities of the modified Benney hierarchy.

A hard line-segment barrier lies in a periodic cell in the plane, parallel to one edge, and half its length. This constitutes perhaps the simplest pseudo-integrable system, and the classical motion of a particle within it admits concise analysis. The hitting and missing of the barrier defines a quasi-periodic (doubly periodic) sequence with the period ratio depending on the invariant (gradient) of the trajectory. The consequent motion is expressed in terms of its continued fraction expansion. Rational gradients correspond to 'periodic' orbits which can be classified in full. Quadratic irrational gradients lead to trajectories of calculable fractal dimension (or power-law correlation decay).

The authors obtain the phase-space quantisation for relativistic spinning particles. The main tool is what they call a 'Stratonovich-Weyl quantiser' which relates functions on phase space to operators on a suitable Hilbert space, and has the essential properties of covariance (under a group representation) and traciality. Their phase spaces are coadjoint orbits of the restricted Poincare group; they compute and explicitly coordinatise the orbits corresponding to massive particles, with or without spin. Some orbits correspond to unitary irreducible representations of the Poincare group; they show that there is a unique Stratonovich-Weyl quantiser from each of these phase spaces to operators on the corresponding representation spaces, and compute it explicitly.

It is shown that the propagator K(xt mod x'0) for a particle in a potential field is derivable from the classical path starting from x' and reaching x at time t. Considering a wavepacket as the particle initial state lying mainly on one side of a barrier the propagator is used for obtaining the wavefunction on the other side. Tunnelling is then discussed in terms of the evolving wavefunction. It is argued that while the particle's expectation energy is taken to be lower than the barrier's height the transformation by which the wavefunction is produced entails energetically crossover flights as required by the dynamics for the classical paths involved in the propagation process. The parabolic repeller exemplifies the formalism and exact results for the probability and current densities are given. A computation involving ballistic tunnelling shows that the current density rises to a saturation value proportional to the particles' injection rate.

The author studies the quantum Lorentz gas with point interactions. The author uses the explicit form of the resolvent of this system to set up an expansion for the pressure and the central limit theorem to sum the many random contributions to the scattering of the light particle due to the heavy particles. The author also discusses the convergence of the expansion.

Dyson's two-dimensional Coulomb gas on the unit circle with inverse temperature beta is investigated with the help of a Metropolis algorithm. Theoretical predictions for energy and specific heat are verified. The connection with the theory of ensembles of random unitary matrices with orthogonal, unitary or symplectic symmetry corresponding to beta =1, 2, 4 is investigated in detail. New approximation formulae for the spacing distribution and the so-called delta statistics are proposed and found to be useful, not only in the three cases mentioned but also for all beta between 0 and 4.

Following the approach introduced by Coniglio et al. (1989) for the Ising model, the author studies the relation between damage spreading and thermal properties in the q-state Potts model as well as in the Ashkin-Teller model. The author shows that by selecting appropriate combinations of different types of damage (q(q-1) in the Potts model and 12 in the Ashkin-Teller model), it is possible to find exact relations between damage and thermodynamic functions, like the magnetisation and the pair correlation function.

An exact relation between one of the recently introduced models for kinetic gelation and random percolation is displayed. This explains the frequently observed identity between the exponents of the two models. Moreover, it strongly suggests that any further universal properties of percolation are also shared by kinetic gelation. In particular, a rigorous inequality due to van den Berg and Kesten (1985) can be used to display an inconsistency between the observed values of the exponents beta and nu and the observed value of the fractal dimension of the backbone in kinetic gelation. The more complex issue of the observed variation of the amplitude ratio cannot be easily resolved, however. Should the observed discrepancies be real, they would, according to the author's results, indicate that the amplitude ratio can indeed vary depending upon the underlying lattice structure.

The authors study the n-component phi ⁴ model in space dimensions 2<or=d<or=4, for various values of n using the Wilson recursion formula. In two dimensions, asymptotic freedom is seen when n>or=3, whereas a phase transition occurs for n=1, with the Ising-type value for the exponent nu . Approaching two-dimensions, the model is shown to tend to the corresponding nonlinear sigma -model S_n-1. The value of the exponent nu compares well with the first-order d-2 expansion. An extension due to Golner (1973), in the scalar case, of the recursion formula and giving an exponent eta not=0, is generalised to the vectorial case. It is solved numerically for 2<or=d<or=4. In three dimensions the value obtained for eta is the same within the error bars as the one computed by field theoretic techniques or high-temperature expansions. The value of eta in the Ising case, when d=2, compares well with the Onsager value.

The Heisenberg relation for the q-analogues of the quantum harmonic oscillator (introduced independently by Macfarlane and Biedenharn (1989) for the quantum group SU(2)_q) is derived by a method analogous to that used by Schwinger (1968) for the SU(2) case. The author also speculates on the possible use of the quantum group as a generalisation of quantum mechanics.

The finite-size scaling spectra of the spin-1 Zamolodchikov-Fateev chain for an even and odd number of sites are presented. The operator content is given for free as well as toroidal boundary conditions. In the case of the antiferromagnetic chain the authors give separately the ferromagnetic and antiferromagnetic excitations.