Table of contents

A method of reducing the partitions of the unitary groups U((n+1)(n+2)/2) to irreducible SU(3) contents is presented.

One-dimensional fluid models with a hard core, a short-range repulsion and a long-range attraction are considered. For certain parameter ranges there is a triple point where vapour, liquid and close-packed phases meet. Results for a lattice model are compared with results for the corresponding continuous model and are found to be qualitatively similar. However, the symmetry with respect to density in the lattice model is absent in the continuous model and the temperature and pressure at the triple point are much higher in the former than in the latter.

The authors give a rigorous treatment of the unitary gauge in the SU(2) Yang-Mills-Higgs system. It is shown that the massive vector fields give a long-range contribution along the negative z axis corresponding to the string in the electromagnetic field.

The authors consider vector-valued functions on a manifold M, taking values in a representation space for a group G which also acts on M, and show how such functions can be resolved into components which transform irreducibly under G, without performing a Clebsch-Gordan decomposition. They also describe the connection with induced representations of G.

It is shown that the bands ( epsilon _m(k) mod m=1,..., infinity ,k in (- pi , pi )) of one-dimensional Bloch Hamiltonians H=pⁿ+V(x), n>or=2, V periodic, are uniquely determined by n-1 fibres ( epsilon _m(k_i) mod m=1,..., infinity , i=1,...,n-1). This extends known results on Hill's equation.

For pt.I see ibid., vol.9, p.11 (1976). When conventional variational trial functions are used to solve systems such as Schrodinger's equation with discontinuous potentials, it is found that the convergence rate is very slow when compared with that for continuous potentials. In a previous paper, the authors demonstrated how this convergence rate could be improved using local variational methods (i.e. finite elements). In this paper they demonstrate the construction of trial functions for global variational methods which results in a very substantial improvement in the obtained convergence rate. The technique, which is numerically stable, is based on orthogonal polynomials with suitable core functions incorporated.

It is shown how an arbitrary number of bound states and resonances with prescribed energies and form factors may be derived from an equivalent separable potential. It is then possible to express the two-body T matrix exactly in terms of the bound state and resonance parameters without explicitly introducing the potential.

Fermi's 'golden rule' expressions refer to first- and second-order time-dependent perturbation theory results for transition probability rates involving designated final states which are members of a continuum. By use of a Laplace-average formalism, it is shown that the long-term time-averaged total transition probability rate involving designated initial and final states which are all members of a continuum is exactly the sum of the pertinent 'golden rule' expressions. Generalization to higher-order perturbation theory results is suggested.

The density of states n(E) is calculated for a bound system whose classical motion is integrable, starting from an expression in terms of the trace of the time-dependent Green function. The novel feature is the use of action-angle variables. This has the advantages that the trace operation reduces to a trivial multiplication and the dependence of n(E) on all classical closed orbits with different topologies appears naturally. The method is contrasted with another, not applicable to integrable systems except in special cases, in which quantization arises from a single closed orbit which is assumed isolated and the trace taken by the method of stationary phase.

The authors study the development of perturbation theory for non-ideal systems of particles. An approach similar to cluster expansions is used. The given perturbation theory can be used when the disturbance of the interaction potential Delta phi is higher than kT but is of a short-range nature.

An n-component spin model, with the nearest-neighbour Hamiltonian H=-J Sigma (S_i.S_j)-K Sigma (S_i.S_j)², where S_i is a discrete unit vector pointing only along one of the 2n cubic axes directions, is studied exactly at dimensions d=1 and d=1+ epsilon and approximately (using dedecoration renormalization group recursion relations) at d=2. The model exhibits four competing possible types of critical behaviour, related to the Ising model, to the n-state and 2n-state Potts models and to a 'new cubic' fixed point. For large n, at d=2, the last three types of behaviour show peculiarities which may be related to the transition becoming a first-order one.

The authors report a microscopic derivation of the parametric representation of the equation of state for an Ising-like system by means of the skeleton expansion technique. Such a representation explicitly describes the crossover behaviour from the small correlation length region to the critical region. The Gaussian-Ising crossover is obtained at criticality. The derivation is based on differential relations that connect natural thermodynamical variables with the parameters defining the representation. Differentiability properties are checked in the small epsilon limit.

The existence of a photon mass implies additional components of static magnetic fields. Schrodinger's expression for a dipole field is here extended to a complete spherical harmonic analysis of a static planetary field. Additional components of the magnetic field are identified which are due to a photon mass. The magnitude of the most important of these new terms is evaluated for the geomagnetic field and is found to be of the same order as Schrodinger's apparent external field.

The problem of electromagnetic wave scattering from arbitrarily shaped scatterers recently solved by using an integral equation formulation is extended to treat the case of scattering by infinite cylinders. The method is based on describing the internal field of the scatterer in terms of an independent set of plane wave functions. Numerical stability conditions are satisfied. Numerical results are presented for cylinders of elliptic cross section for two independent incident waves. Results are also given for scattering by three-dimensional spheroidal scatterers of refractive index n₀=1.33. The solution presented is applicable for scatterers with size parameter (k₀a) from the Rayleigh region (k₀a<<1) up to the geometrical optics limit (k₀a>>1) provided for each given scatterer certain matrix elements can be calculated.

A systematic method to treat the steady propagation of a coherent light pulse in a dielectric medium is developed, paying special attention to the effect of polariton formation. The nonlinear optical Bloch equation and the Maxwell equation are solved simultaneously by means of the method of series expansion in powers of a small parameter which is related to the pulse width. It is shown that, besides the usual self-induced transparency pulse, steady pulse solutions exist also in the case where the pulse width is much longer than the reciprocal of the polariton gap frequency. This result suggests that steady propagation may be realized also in dense dielectric media such as crystals in which the polariton effect is marked.

The time structure of atmospheric Cerenkov light has been measured by narrow-angle detectors in extensive air showers with a time resolution of 0.15 ns. From comparison with particle density observations in the same showers it is deduced that the temporal shape is a measure of the lateral structure of the particle shower disc rather than of the longitudinal shower development.