Table of contents

Dual transformations in two-dimensional classical and quantum mechanical systems have been widely studied using conformal mapping techniques but one-dimensional systems have been largely ignored. In this paper we study dual transformations in one-dimensional mechanical systems, both classical and quantum mechanical, using some previously developed methods. A number of examples, mostly involving periodic motion or bound states, are presented. Dual transformations provide interesting connections between hitherto unconnected problems.

The exact amplitude for the asymptotic correlation function in the Heisenberg antiferromagnetic chain is determined:

The behaviour of the correlation functions for small xxz anisotropy and the form of finite-size corrections to the correlation function are also analysed.

Attractors in asymmetric neural networks with deterministic parallel dynamics present a `chaotic' regime at symmetry where the average length of the cycles increases exponentially with system size, and an oscillatory regime at high symmetry, where the average length of the cycles is 2. We show, both with analytic arguments and numerically, that there is a sharp transition, at a critical symmetry , between a phase where the typical cycles have length 2 and basins of attraction of vanishing weight and a phase where the typical cycles are exponentially long with system size, and the weights of their attraction basins are distributed as in a random map with reversal symmetry. The timescale after which cycles are reached grows exponentially with system size N, and the exponent vanishes in the symmetric limit, where . The transition can be related to the dynamics of the infinite system (where cycles are never reached), using the closing probabilities as a tool.

We also study the relaxation of the function , where is the local field experienced by the neuron i. In the symmetric system, it plays the role of a Ljapunov function which drives the system towards its minima through steepest descent. This interpretation survives, even if only on the average, also for small asymmetry. This acts like an effective temperature: the larger the asymmetry, the faster the relaxation, and the higher the asymptotic value reached. E reaches very deep minima in the fixed points of the dynamics, which are reached with vanishing probability, and attains a larger value on the typical attractors, which are cycles of length 2.

We study the stability of the O(N) fixed point in three dimensions under perturbations of the cubic type. We address this problem in the three cases N = 2,3,4 by using finite-size scaling techniques and high-precision Monte Carlo simulations. It is well known that there is a critical value below which the O(N) fixed point is stable and above which the cubic fixed point becomes the stable one. Whilst we cannot exclude that , as recently claimed, our analysis strongly suggests that coincides with 3.

We propose an integrable open XXZ chain coupled to the boundary impurities with arbitrary exchange constants. The Bethe ansatz equation and eigenvalues are obtained by using the quantum inverse scattering method. The ground-state properties are discussed by solving the Bethe equation in some special cases. In addition, we present an approach for constructing the reflecting matrix K which produces the boundary term coupled to impurities.

The optimization of measurement for n samples of pure states are studied. The error of the optimal measurement for n samples is asymptotically compared with the one of the maximum likelihood estimators from n data given by the optimal measurement for one sample.

By means of extensive computer simulations we analyse in detail the two-dimensional Ising spin glass with ferromagnetic next-nearest-neighbour interactions. We found a crossover from ferromagnetic to `spin glass'-like order both from numerical simulations and analytical arguments. We also present proof of a second crossover from the `spin glass' behaviour to a paramagnetic phase for the largest volume studied.

This work is an extensive study of the spectral statistics of three representative classically integrable systems, namely rectangle, torus and circle billiards. We analyse the statistics and focus on the related level spacing distribution P(S) and the delta statistics . The agreement with the Poissonian model is typically found to be perfect up to the outer (unfolded) energy scale , beyond which the saturation is observed, in agreement with Berry's dynamical theory of the spectral rigidity, where as .

The untypical systems are those, where for example P(S) is not a smooth distribution but a sum of the delta functions, due to the `granularity' of the energy scale, for example in the rectangle with the rational squared sides ratio. However, even there we find reasonable trend towards Poissonian statistics for large ranges L but . We describe theoretically and numerically the broadening of the delta spikes when the rectangular billiard is slightly distorted away from a rational to an irrational shape and find excellent agreement. Also, in irrational rectangle billiards we show and explain the existence of large fluctuations, by one order of magnitude bigger than the statistical ones, whose origin is in the closeness to some rational billiard shape. These fluctuations and their amplitude are independent of the energy if the bin size shrinks inversely with energy.

Finally we tested the mode distribution (i.e. distribution of the reduced mode fluctuation number W) and found that it was not Poissonian, in agreement with Steiner's conjecture, and in fact follows the prediction by Bleher et al, that its tail behaves as . The general reason for non-universal behaviour of P(W) is that at largest energy scales we are always in the saturation regime .

A generic su(1,1) Tavis-Cummings model is solved both by the quantum inverse method and within a conventional quantum-mechanical approach. Examples of corresponding quantum dynamics including squeezing properties of the su(1,1) Perelomov coherent states for the multiatom case are given.

We present analyses of substantially extended series for both interacting self-avoiding walks (ISAW) and polygons (ISAP) on the square lattice. We argue that these provide good evidence that the free energies of both linear and ring polymers are equal above the -temperature, thus extending the application of a theorem of Tesi et al to two dimensions. Below the -temperature the conditions of this theorem break down, in contradistinction to three dimensions, but an analysis of the ratio of the partition functions for ISAP and ISAW indicates that the free energies are in fact equal at all temperatures within at least. Any perceived difference can be interpreted as the difference in the size of corrections to scaling in both problems. This may be used to explain the vastly different values of the crossover exponent previously estimated for ISAP to that predicted theoretically, and numerically confirmed, for ISAW. An analysis of newly extended neighbour-avoiding self-avoiding walk series is also given.

A direct method to derive the expression for the polarization potential between a charged particle and a two-fragment quantum system is developed. This method is based on the asymptotic properties of the two-body Coulomb Green function. The explicit form of the polarization potential constant is obtained for low-energy deuteron-nucleus scattering and for ion-hydrogen scattering. The properties of the polarization potential at high collision energies are applied to obtain an expression for the ion-atom scattering total cross section.

The so-called Belov-Chaltikian lattice is considered. By the dependent variable transformation, the Belov-Chaltikian lattice is transformed into a trilinear form. By introducing an auxiliary variable, we further transform it into the bilinear form. A corresponding Bäcklund transformation for it is obtained. Furthermore, a nonlinear superposition formula is proved rigorously. As an application of the obtained results, soliton solutions are derived.

We construct a unified recurrence operator method for obtaining explicit expressions for the wavefunctions of shape-invariant potentials. It is found that the normalized coefficients for the energy eigenfunctions satisfy a universal recurrence relation. The procedure is illustrated in detail for four potentials. We work out the normalized explicit wavefunctions of Hulthen potential, for which the normalized explicit wavefunctions have not been previously calculated.

We study random dilution of random matrices , where are uniformly distributed over the group of unitary matrices and are non-random Hermitian matrices. We show that the eigenvalue distribution function of dilute random matrices converges to the semicircle (Wigner) distribution in the limit , , where p is the dilution parameter. This convergence can be explained by the observation that the dilution eliminates statistical dependence between the entries of . The same statement is valid for the entries of . Our results support the conjecture that the Wigner law is valid for wide classes of dilute Hermitian random matrices.

The path integral for the relativistic spinless Aharonov-Bohm-Coulomb system is performed. The energy spectra and wavefunctions are extracted from the resulting amplitude.

In this paper we first characterize the Lie algebra of derivations of the three-dimensional Manin quantum space as the semi-direct product of the Lie algebra of its inner derivations and the threefold generalized Virasoro algebra with central charge zero. Then we consider Hamiltonian systems on the quantum plane and we prove that the set of Hamiltonian derivations is a Virasoro algebra with central charge zero. Moreover, we show that the only possible motions on the quantum plane come from quadratic Hamiltonians and we find the solutions of the corresponding Hamilton equations explicitly.

The gauge equivalence between the Manin-Radul and Laberge-Mathieu super KdV hierarchies is revisited. Apart from the Inami-Kanno transformation, we show that there is another gauge transformation which also possesses the canonical property. We explore the relationship of these two gauge transformations from the Kupershmidt-Wilson theorem viewpoint and, as a by-product, obtain the Darboux-Bäcklund transformation for the Manin-Radul super KdV hierarchy. The geometrical intepretation of these transformations is also briefly discussed.

For quantum systems with two-dimensional configuration space we construct a physical radial momentum observable. Rescaling the radius we find that the dilation degrees of freedom form a Weyl algebra. With this we construct a radial Wigner quasiprobability distribution function.