Table of contents

Based on the Temperley-Lieb algebra we define a class of one-dimensional Hamiltonians with nearest- and next-nearest-neighbour interactions. Using the regular representation we present ground states of this model as words of the algebra. Two-point correlation functions can be computed employing the Temperley-Lieb relations. Choosing a spin- 1/2 representation of the algebra we obtain a generalization of the (q-deformed) Majumdar-Ghosh model. The ground states become valence-bond states.

The boson-fermion correspondence is applied to derive explicit formulae for expressing the product of S-functions in terms of sums of S-functions associated to non-standard partitions.

The effects of symmetry breaking and renormalization on a unified field theory can be simulated by the use of a deformed gauge group instead of U(3). Here the use of a deformation of u(3) called u_Q(3), with three independent parameters, is proposed and illustrated by obtaining a mass formula with electromagnetic mass splitting as well as the hypercharge and isospin dependence for the baryons.

We discuss a dynamical model in which Dirac's constraint theory generates a continuum of possible dynamics algebras. Upon canonical quantization almost all are found to violate the Jacobi identity. Of the two anomaly-free algebras, one is Snyder's quantized spacetime algebra and the other is a curved momentum space algebra.

The recently developed quantum surface-of-section method is applied to a search for extremely high-lying energy levels in a simple but generic Hamiltonian system between integrability and chaos, namely the semiseparable two-dimensional oscillator. Using the stretch of 13445 consecutive levels with the sequential number around 1.8*10⁷ (eighteen million) we have clearly demonstrated the validity of the semiclassical Berry-Robnik level-spacing distribution while at 1000 times smaller sequential quantum numbers we find the very persistent quasi-universal phenomenon of power-law level repulsion which is globally very well described by the phenomenological Brody distribution. This is the first statistically significant numerical confirmation of the Berry-Robnik surmise in an autonomous two-dimensional Hamiltonian system.

The Hopf structure of the central extension of the U_q(sl(n)) algebra is considered. The intertwine matrix induces new integrable chain models. We show the relation of these models and the biparametric spin chain sl_p,q(n) models. The cases n=2 are n=3 are discussed, and for n=2 we obtain the model of Dasgupta and Chowdhury (1993). The case n=3 is solved by the nested Bethe ansatz method, and the dependence of the Bethe equations on the second parameter introduced is demonstrated.

A physically natural generalization of the notion of observable that encompasses both the classical and the quantum ones is derived. Based on it, the idea of the classical extension of a theory is developed; the states of the extended theory being the probability measures on the pure states of the original one. It is shown that quantum theory admits such a classical extension and that the qualifying features of quantum observables are preserved in the extended model.

The Privman-Fisher universal amplitude, characterizing the free energy of a finite-sized system at its critical point, is identified as the critical counterpart of the 'number of stable phases' on the coexistence curve. Its value is shown to be determined by the critical-point form of the order parameter distribution, directly accessible to Monte Carlo studies.

The finite-size corrections for the anisotropic Heisenberg model in the gap region are calculated using an exact integral representation based on the Bethe ansatz equations (BAE). The energy corrections are determined by saddle-point approximations for states without complex roots. The case of symmetric states is discussed in greater detail.

Explicit formulae for the calculation of traces of N-order replacement operators in finite-dimensional, antisymmetric and spin-adapted N-electron Hilbert spaces are derived. These formulae are useful in the calculation of traces of Hamiltonian powers and provide an easier determination of moments of spectral density distributions according to the general procedure reported recently.

We show that in frustrated spin-S Ising antiferromagnets the entropy can stabilize a partially-ordered ground state. In the model on the decorated square lattice partial long-range order exists for S>or=1. For the model on the union jack lattice the partially-ordered ground state exists even for S=1/2. For this lattice the ground-state problem is shown to be equivalent in a certain case to the staggered six-vertex model. We suggest that the model on the triangular lattice for S>S_C approximately 3 might exhibit a novel kind of partial ordering.

We compare the capabilities of different nonlinear algorithms for forecasting chaotic time series when a limited number of past values of the series is available, a situation most often found in real-world problems. In particular, we consider instance-based methods and neural network techniques, which are frequently advocated in the literature as universal, simple, and fairly reliable algorithms for time-series analysis. Furthermore, we propose a linear correction to the instance-based Wimplex method that produces remarkably good results on clean data. Finally, we present a preliminary application of the ideas discussed here to the real-world series of solar activity, which is often taken as a benchmark for these kinds of studies.

The fermionic version of the two-dimensional Ising model in a uniform magnetic field is studied. Its approximate but rather accurate solution was obtained recently in a variational procedure based on the BCS ansatz for the ground state. Here, this solution is continued analytically into the complex H-plane in the vicinity of the origin. In the ferromagnetic phase an essential singularity in the free-energy is observed at the point H=0 of the type predicted by the droplet condensation theory. Simple phenomenological estimates for the radius of the critical droplet and for the rate of decay of metastable states are confirmed by microscopic calculations.

We study the effect of an external driving force on a simple stochastic reaction-diffusion system in one dimension. In our model each lattice site may be occupied by at most one particle. These particles hop with rates (1+or- eta )/2 to the right and left nearest neighbouring site respectively if this site is vacant, and annihilate with rate one if it is occupied. We show that density fluctuations (i.e. the mth moments (N^m) of the density distribution at time t) do not depend on the spatial anisotropy eta induced by the driving field, irrespective of the initial condition. Furthermore, we show that if one takes certain translationally invariant averages over initial states (e.g. random initial conditions) even local fluctuations do not depend on eta . In the scaling regime t approximately L² the effect of the driving can be completely absorbed in a Galilei transformation. We compute the probability of finding a system of L sites in its stationary state at time t if it was fully occupied at time t₀=0.

In this paper we show the existence of eta -pairing and off-diagonal long-range order (ODLRO) in the ground state of an extended Hubbard model with N up arrow -N down arrow =1, where N up arrow and N down arrow are the occupation numbers of up and down spins, for connected bipartite lattices.

We investigate the effects on a network that stores a sequence of patterns without time delays of introducing patterns that are correlated in a way that depends only on their separation from each other in the sequence. We demonstrate that with a carefully chosen form for the synaptic matrix, the introduction of these correlations can, under certain circumstances, aid the recall of the individual patterns while still preserving the limit-cycle behaviour of the net.

We investigate a quantum two-oscillator model for second-harmonic generation. The total Hamiltonian is the sum of two commuting Hamiltonians with eigenvalues E₀ and E_int. The exact determination of these eigenvalues is studied using tridiagonal matrices. We present two general equations in the quasiclassical regime yielding the largest E_int for a given E₀ and a representation for both eigenvalues in terms of an additional quantum number. Some numerical analysis shows that both equations are fairly consistent for suitably large quantum numbers with the exact quantum results. An approximate analytical expression for the wavefunction is also given.

The existence of a class of exact solutions for diffusion-reaction problems is demonstrated for the example of a band. The method may be applied to any particle source which can be analytically mapped to a band (or a disc) situated on an infinite plane. A simple asymptotic expression for the experimentally measurable collector efficiency (current-current response function) has been found.

Transition state theory as developed for chemical reactions is transferred to ballistic electrons. The phase-space flux through the transition state is related to the conductance. Steps in the conductance follow from a semiclassical extension of this formula. Tunnelling corrections have to be included to yield the rounding of the steps. A magnetic field causes a sharpening of the steps and at finite temperature an additional broadening. The calculation also shed some light on the abilities and limitations of the semiclassical approach.

We consider the recently obtained integral representation of the quantum Knizhnik-Zamolodchikov equation of level 0. We obtain the condition for the integral kernel such that these solutions satisfy three axioms for the form factor in the manner of Smirnov. We discuss the relation between this integral representation and the form factor of the XXZ spin chain.

The scattering problem related to the Schrodinger operator with an external constant homogeneous magnetic field is considered. The behaviour of the corresponding S-matrix and resolvent operator near the Landau thresholds is studied. The large-energy asymptotics of the resolvent operator are evaluated. A Levinson-type formula is obtained. It relates the number of bound states to the determinant of the scattering matrix in the scattering threshold.

Integrable dynamical systems are constructed on semi-direct non-compact Lie algebra SO(6)* nu ₆, where nu ₆ is the topological vector space on which the representation of SO(6) is realized. The construction is done with the help of the sectional operators introduced by Trofimov and Fomenko (1983). Our equations generalize those of a rigid body immersed in a fluid and those of Manakov (1976) on SO(6). Our equations contain the system of two interacting rigid bodies immersed in a fluid, as a special case.

A method to generate isospectral supersymmetric Hamiltonians is given which is similar but inequivalent to the usual approach based on Darboux's theorem. Explicit calculations are made for the potential V(r)=-2 sech² r and the conditionally exactly solvable potential given by V(r)=Ar²3/+B/r²3/-G₀/r².

It is shown that from some solutions of generalized Knizhnik-Zamolodchikov equations one can construct eigenfunctions of the Calogero-Sutherland-Moser Hamiltonians with exchange terms, which are characterized by any given permutational symmetry under particle exchange. This generalizes some results previously derived by Matsuo (1992) and Cherednik (1991) for the ordinary Calogero-Sutherland-Moser Hamiltonians.

By analogy to the continuous Painleve-II equation, we present particular solutions of the discrete Painleve-II (d-P_II) equation. These solutions are of a rational and special function (Airy) type. Our analysis is based on the bilinear formalism that allows us to obtain the tau -function for d-P_II. Two different forms of bilinear d-P_II are obtained and we show that they can be related by a simple gauge transformation.

We introduce the general deformed differential operator. Using it the general deformed KdV hierarchy is constructed. As an elementary example, we give the general deformed KdV equations. In the form of the general deformation parameter expansion the general deformed KdV equations generates the systems of the non-homogeneous linearized KdV equations which are exactly solvable. In order to solve the general deformed KdV hierarchy, we introduce the general deformed exponential function. By using the dressed-operator method these exact solutions can be constructed. As an elementary example, we give some exact solutions of the non-homogeneous linearized KdV equation system derived from the general deformed KdV equations.

We comment on our paper 'Exact solutions for the coagulation-fragmentation equation' (ibid., vol.25, p.4737, 1992).