Table of contents

A generalized Hirota's bilinear equation is considered. Furthermore, two special forms of it are studied in some detail. Under certain conditions, the author shows that the equations have one- or two-soliton solutions. The conditions under which three-soliton or N-soliton solutions exist are also given. Some examples are illustrated.

The spectrum of a one-dimensional chain of SU(n) spins positioned at the static equilibrium positions of the particles in the corresponding classical Calogero system with an exchange interaction inversely proportional to the square of their distance is studied. As in the translationally invariant Haldane-Shastry model the spectrum is found to exhibit a very simple structure containing highly degenerate 'super-multiplets'. The algebra underlying this structure is identified and several sets of raising and lowering operators are given explicitly. On the basis of this algebra and numerical studies the authors give the complete spectrum and thermodynamics of the SU(2) system.

The authors have studied the branched polymer model, situated on the Sierpinski gasket lattice, with competing monomer-monomer and monomer-surface interactions. They have determined critical properties of this model within an exact renormalization group approach. Their results reveal that the corresponding model system possesses a rich phase diagram that is typical for polymer surface physics.

The author exactly calculates the anisotropic interfacial tension of a square lattice gas model with nearest-neighbour exclusion and next-nearest-neighbour attractions. This is achieved by a method which introduces the shift operator into the standard transfer matrix argument. From the calculated anisotropic interfacial tension the equilibrium crystal shape (ECS) is derived by the use of the Wulff construction. The ECS is a closed curve in the X-Y plane, represented as cosh( Lambda (X+Y)/k_BT)+A₃cosh( Lambda (X-Y)/k_BT)+A₄/2=0 with a scale factor Lambda and constants A₃,A₄. He argues that this shape is a universal one which appears as the ECSS of a wide class of models.

An equivalence between generalized restricted solid-on-solid models, associated with sets of graphs, and multi-colour loop models is established. As an application the authors consider solvable loop models and, in this way, obtain new solvable families of critical RSOS models. These families can all be classified by the Dynkin diagrams of the simply laced Lie algebras. For one of the RSOS models, labelled by the Lie algebra pair (A_L,A_L) and related to the C₂⁽¹⁾ vertex model, they give an off-critical extension, which breaks the Z₂ symmetry of the Dynkin diagrams.

The properties of a family of nonequilibrium spin models with up-down symmetry on a square lattice are determined by a mean-field pair approximation and by Monte Carlo simulation. The phase diagram in the parameter space displays a critical line that terminates at a first-order critical point. It is found that the critical exponents are the same as those of the equilibrium Ising model.

The author shows that the correlation function of a local operator B_i decays if there is another local operator A_i satisfying (H,A_i)= alpha B_i, where H is the Hamiltonian of the many-body system under consideration and alpha is a constant. Finally, as an application of this theorem, he shall rigorously show that the RVB states, which were proposed by P. W. Anderson (1987) and his collaborators to explain high-temperature superconductivity, are absent in the Hubbard model at half-filling. He also gives an argument, which indicates that the existence of the RVB ground states in the doped cases is highly improbable.

A mean-field analysis is presented of the retrieval behaviour of a modular network of binary neurons capable of storing and retrieving patterns made up of associated features.

A model describing the dynamics of the synaptic weights of a single neuron performing Hebbian learning is described. The neutron is repeatedly excited by a set of input patterns. Its response is modelled as a continuous, nonlinear function of its excitation. The authors study how the model forms a self-organized representation of the set of input patterns. The dynamical equations ae solved directly in a few simple cases. The model is studied for random patterns by a signal-to-noise analysis and by introducing a partition function and applying the replica approach. As the number of patterns is increased a first-order phase transition occurs where the neuron becomes unable to remember one pattern but learns instead a mixture of very many patterns. The critical number of patterns for this transition scales as N^b, where N is the number of synapses and b is the degree of nonlinearity. The leading order finite-size corrections are calculated and compared with numerical simulations. It is shown how the representation of the input patterns learned by the neutron depends upon the nonlinearity in the neuron's response. Two types of behaviour can be identified depending on the degree of nonlinearity; either the neuron learns to discriminate one pattern from all the others, or it will learn to discriminate a complex mixture of many of the patterns.

A generic one-parameter family of billiards discovered and introduced by Robnik (1983) is used to study the spectral properties of corresponding quantum systems. When the parameter is varied a smooth transition from an integrable system over a typical KAM system to an almost ergodic system can be observed. The authors calculate up to 7600 lowest reliable energy levels. A detailed analysis of the numerical data shows significant deviation from the semiclassical Berry-Robnik formulae for the nearest-neighbour level spacing distribution P(S) except for large level spacings, S>1, which can only be explained by a very slow convergence towards the semiclassical regime where these formulae are predicted to be correct. At small S the power-law level repulsion is clearly observed and a fit by the phenomenological formula by Izrailev (1988,1989) is statistically significant.

A reasonable q-deformed differential is defined. A set of operation rules are constructed for the q-deformed formal pseudo differential operators. The complete procedure of constructing the q-deformed KdV hierarchies and their infinite conservation laws is given. As an important example, he obtains a detailed structure of the simplest (3,2) system, i.e. the q-deformed KdV equations.

The authors study the sum zeta _H(s)= Sigma _jE_j^-s over the eigenvalues E_j of the Schrodinger equation in a spherical domain with Dirichlet walls, threaded by a line of magnetic flux. Rather than using Green function techniques, they tackle the mathematically non-trivial problem of finding exact sum rules for the zeros of Bessel functions J_v, which are extremely helpful when seeking numerical approximations to ground state energies. These results are particularly valuable if v is neither an integer nor half an odd integer.

The author considers an Abelian gauge field theory on Euclidean partially compactified spacetime T^N*Rⁿ for arbitrary N and n. The one-loop effective potential generated by quantum fluctuations of a massive scalar field, minimally coupled to a constant background Abelian gauge potential is calculated for arbitrary compactification lengths L₁,. . .,L_N of the multi-dimensional torus. In particular the topologically generated mass of the gauge field is obtained and its complicated dependence on the parameters involved (compactification lengths, mass M of the scalar field) is given explicitly. It is found that the topologically generated mass is positive for arbitrary N and n and that it does not depend on a renormalization parameter. For n>2 the limit M to 0 is smooth, but for n=0,1,2 zero modes play a crucial role and the generation of real or imaginary gauge field masses is possible.

The authors prove the existence and show how to construct trace maps for products of 2*2 matrices generated by arbitrary substitution sequences. The dimension of the underlying space of the trace map is lower than the one suggested recently by other authors.

A few years ago the author and his collaborators introduced the concept of a Dirac oscillator and extended it to a quark-antiquark (qq) system for a discussion of the mass spectrum of mesons. The problem was reduced to a radial equation of a familiar type but with a singularity (r-a)^-1 identical to x^-1 at some given value of the radius. The character of the mass spectrum was determined by whether the potential x^-1 with - infinity <or=x<or= infinity is penetrable at the origin or not. This leads to a discussion of penetrability of a one-dimensional Coulomb barrier which is the object of this work. The solution of the corresponding wave equation for x<0 and x>0 is well known, but the trick is to join them at x=0 where they are bounded but have an infinity derivative. The authors obtain explicitly the transmission and reflection amplitudes. As the barrier is then penetrable, the qq system does not have a bound spectrum, but a continuous one which, in some cases, may have resonances whose widths are small compared with their separation.