Table of contents

Completes the author's investigation (see Phys. Lett.A, vol.81, p.17-8, 1981) and presents exact solutions and eigenvalues for anharmonic oscillators, the solutions being given in the form of definite integrals. The conditions for the validity of these results are investigated.

The complex index of gravitational wave refraction in condensed matter systems is related to the viscosity response function. Simple applications are discussed for gravitational wave propagation in fluids and crystals.

Results of computer simulations for the antiferromagnetic q-state Potts model on the triangle and FCC lattices are reported. For the triangle lattice, the three-state Potts model has a first-order phase transition at a temperature T_c approximately=0.63 mod J mod . For q>or=4, the system is paramagnetic for all temperatures. For the FCC lattice, the transition is first order for 2<or=q<or=6.

The behaviour of an interface subject to an external pinning forces( is studied in a molecular-field theory. In the case of a pinning force applied near the edge of the system, one finds a finite-temperature localisation-delocalisation transition in the solid-on-solid (SOS) model but not in the Gaussian model.

The general form of partial differential equations integrable by the arbitrary-order linear spectral problem is found. The groups of Backlund transformations corresponding to these equations are constructed. It is shown that partial differential equations of the class under study are Hamiltonian ones. Some reductions of general equations are considered. In particular, the Hamiltonian structure of the generalisations of the sine-Gordon equation to the groups GL(N), SU(N), and SO(N) at arbitrary N is proved.

By using the calculus on complex Banach spaces developed by Sharma and Rebelo (1975), the authors give a rigorous proof of the simplest and yet the most generalised version of the Brillouin theorem as given by Sharma and SriRankanathan (1980).

Optimal constraints on the values of a physical amplitude are derived when its imaginary part and an upper bound on the modulus are given along distinct regions of the boundary of the analyticity domain. The problem is formulated rigorously as a minimum norm problem in a space of analytic functions and solved by applying a duality theorem. An approximate description of the solution, suitable for practical applications, is presented.

Isotropic tensors play an important role in the theory of many physical processes, which take place in gases and liquids. In such systems it is usually necessary to perform a rotational average on products of direction cosines relating the space-fixed and molecular coordinate frames. The average is generally expressible in terms of isotropic tensor. The authors discuss the isotropic tensors of eighth rank, and relations between them are demonstrated. The rotational average of eighth rank is then evaluated in both reducible and irreducible form; the results are applicable to a number of processes, for example optical seventh harmonic generation and four-photon absorption.

The ground state wavefunction of an infinitely large, spatially homogeneous system is evaluated in terms of wavefunctions for linked clusters of points within the system. The linked cluster wavefunction is shown to obey a nonlinear form of the Schrodinger equation, with the unlinked pieces providing the nonlinearity. A variational principle for the ground state energy is derived.

Within the framework of the Kershaw stochastic model (1964) and of the hypothesis on the space stochasticity, equations of motion for two relativistic stochastic particles are obtained which coincide in form with the equations of Cufaro Petroni and Vigier (1979). In the nonrelativistic limit, these equations of free motion reduce to the usual two-particle Schrodinger equation, the imaginary part of which corresponds to a Hamilton-Jacobi equation for two particles interacting through a nonlocal quantum potential.

Presents the quantum field theory of the free scalar massless field and derives on this basis the operator solution to the Thirring model. The Lorentz and scale transformation formulae for the scalar and Thirring fields are found explicitly. It is shown that one can attribute definite spin and scale dimension to the Thirring field psi (x) only if one interprets psi (x) as an intertwining operator between inequivalent charged sectors. Particular attention is paid to the way the two charge operators are contained in the theory of the massless scalar field.

The kinematical description of a classical system having a linear phase space can be quantised if a complexification is given over the symplectic space that supports the description; indeed, a complexification determines a Segal quantisation. The question of the uniqueness of the Segal quantisation arises since the quantum descriptions constructed through different Segal quantisations need not be equivalent. If the classical description contains also a symmetry group (represented by linear symplectic transformations; this means that one deals with systems having linear dynamics only), the complexification to determine a Segal quantisation that also quantises the symmetry group. The authors show that this requirement determines uniquely the complexification, and therefore the Segal quantisation, provided the symmetry group fulfils a (real) irreducibility condition. They apply this uniqueness criterion to the free neutral scalar field.

It is shown that, in a static vacuum space-time, the dominant term of the Riemann tensor near a normal-dominated singularity is in general of Petrov type I, although for certain values of a parameter gamma that occurs in the metric it is of type D.

The behaviour of long unscreened polyelectrolyte chains is studied using field-theoretic renormalisation group methods based on the renormalisation procedure of t'Hooft and Veltman (1972). It is shown that a fixed point exists near dimension six and details are given of the calculation of the critical exponents up to O( epsilon ²). It is also shown exactly that, as long as a fixed point exists, the radius of gyration R_G is given by R_G approximately N^v with nu =2/(d-2).

Extends studies of the quantised fermion-boson model. Parallel numeric and approximate analytic advances allow greater understanding of the 'collapse' regions between successive 'revivals' in the fermion energy evolution. The authors give an estimate of the time for the onset of continuous irregularity as a function of the average boson number.

Considers the magnetic response of a charged Brownian particle undergoing a stochastic birth-death process. The latter simulates the electron-hole pair production and recombination in semiconductors. The authors obtain non-zero, orbital diamagnetism which can be large without violating the Van Leeuwen theorem (1921).

The critical behaviour of a general discrete one-dimensional model with inverse square interactions is discussed, using renormalisation group methods.

Hamiltonians in one space dimension of the phi -four and sine-Gordon classes are considered, emphasising kink- and breather-like excitations. A calculation of classical dynamic structure factors based on a kink ideal gas phenomenology is reviewed in terms of the 'central peaks' predicted. This phenomenological scheme is extended to include breather excitations. It is suggested that, for correlations of appropriate functions, the breather excitations can give rise to a low-frequency ('central') response from their particle-like envelope and a high-frequency response from their internal oscillatory motions in qualitative accord with molecular dynamics simulations.

Defines displaced Fock representations of the canonical commutation relations in an algebraic framework. Then the author considers the problem of unitary implementability of the symmetry group which acts on the one-particle space of the theory. This action induces an automorphism of the algebra of the CCR and a condition is found to ensure that this automorphism be implemented by a unitary group representation in the space of the displaced Fock representation. Using this condition the author proves the existence of infinitely many displaced Fock representations in which the group automorphism is implemented by a unitary action. This is done for a specific choice of a subgroup of the Poincare group, and for all cases of integer value of spin.

A certain form for the self-dual solutions of the Yang-Mills-Higgs system is tested. (When restricted to spherical symmetry this ansatz is the most general form in the Wigner-Eckart sense.) It is found that the only consistent solutions of this form are spherically symmetric, and that the only finite energy solution then is the Prasad-Sommerfield solution (1975).

The self-dual SU(3) Yang-Mills equations in Yang's R-gauge (1977) are expressed geometrically as a harmonic map of Riemannian manifolds. The study of geodesics and isometries in the manifold of fields provides certain new families of solutions of the Yang-Mills equations and a continuous group of transformations generating, from any family of solutions, a larger family with two additional parameters.

Makes extensive use of the Mellin transform technique to effect the Fourier-Bessel transformation between the impact parameter and transverse momentum space formulations of the Drell-Yan process for q_T²<<q², thereby demonstrating the complete equivalence between the two schemes.

The Hamiltonian of a non-relativistic free charged particle interacting with one quantised mode of the electromagnetic field is diagonalised exactly in dipole approximation. The diagonalisation is simpler for a circularly polarised mode, but is is shown that the case of a linearly polarised mode can be reduced to the circularly polarised one. Exact stationary and time-dependent states of this Hamiltonian are then calculated. Based on these results, the rederivation of the nonlinear inverse and direct bremsstrahlung's cross section is presented and the connection with earlier (semiclassical) approaches is also established. Photon number distribution functions for different initial conditions are also determined in analytical form.

One of the interesting features of the RISM approximation for molecular fluids is that the results of the theory show dependence on the presence or absence of auxiliary (i.e. non-interacting) sites. The authors examine this dependence analytically and numerically. Using graph-theoretical methods the diagrammatic expansions for the site-site correlation functions are examined in great detail in order to explain the way in which the dependence of auxiliary sites arises in the RISM approximation. This provides considerable insight into the nature of the RISM approximation itself.

The motion of a quantum particle in a one-dimensional, slowly varying periodical potential is considered. By using the comparison equation method, the asymptotic expressions for exponentially small bandwidths of the low-lying energy bands is obtained. In the special case of a sinusoidal potential (corresponding to the Mathieu differential equation), the general formula reduces to an already existing result. Potentials, symmetrical and non-symmetrical, within the elementary period are discussed.