Table of contents

We study the spectral statistics for quantized skew translations on the torus, which are ergodic but not mixing for irrational parameters. It is shown explicitly that in this case the level-spacing distribution and other common spectral statistics, like the number variance, do not exist in the semiclassical limit.

A field-theoretic description of the critical behaviour of systems with quenched defects obeying power law correlations ~|x|^-a for large separations x is given. Directly, for three-dimensional systems and for different values of the correlation parameter, 2a3, a renormalization analysis of the scaling function in the two-loop approximation is carried out, and the fixed points corresponding to the stability of various types of critical behaviour are identified. The obtained results essentially differ from results evaluated by a double ,-expansion. The critical exponents in the two-loop approximation are calculated with the use of the Padé-Borel summation technique.

A phase diagram for a surface-interacting long flexible polymer chain in a poor solvent where the possibility of collapse exists is calculated using an exact enumeration method. A model of a self-attracting self-avoiding walk on a simple cubic lattice was considered and up to 16 steps in series were evaluated. The phase diagram indicates that while the boundary between the expanded and collapsed phases is straight in the bulk, it exhibits a bend in the surface resulting in two adsorbed collapsed phases separated by an adsorbed expanded phase. This is attributed to competition between the entropic fluctuations and effects due to monomer-monomer attraction.

We present a Monte Carlo study of the two-component ⁴ model on the simple cubic lattice in three dimensions. By suitable tuning of the coupling constant, , we eliminate leading-order corrections to scaling. High-statistics simulations using finite-size scaling techniques yield = 0.6723(3)[8] and = 0.0381(2)[2], where the statistical and systematical errors are given in the first and second bracket, respectively. These results are more precise than any previous theoretical estimate of the critical exponents for the 3D XY universality class.

The scalar curvature (R) of ideal quantum gases obeying Gentile's statistics is investigated by the method of information geometrical theory. The R value is specified by the fugacity and the maximum number, p, of particles in a state. The lowest case p = 1, corresponds to Fermi-Dirac statistics and the unbounded case, p , to Bose-Einstein statistics. In contrast to R = 0 for ideal classical gases obeying Boltzmann statistics, we find R = (2)^1/2/32 for p2 and R = -(2)^1/2/32 for p = 1, in 0 which is the classical limit. This means that a quantum statistical character is left in R, in the classical limit. Also, a correlation between the sign of R and a quantum mechanical exchange effect is recognized for 0 and >>1. Furthermore, we obtain results that support the instability interpretation of R proposed by Janyszek and Mrugala.

We present a very simple and efficient method to obtain the exact wavefunction corresponding to the harmonic oscillator with time-dependent mass and frequency.

Based on the phase-space generating functional of the Green function for a constrained Hamiltonian system with a singular higher-order Lagrangian, the canonical Ward identities for such a system under the local and global transformation have been derived, respectively. The quantal conserved charge (QCC) under the global symmetry transformation is also deduced. In general, these QCCs are different from the Noether charge in classical theory. A comparison of these quantal conservation laws with those deriving from the configuration-space path integral for gauge-invariant theories is discussed. We give a preliminary application of our results to Yang-Mills (YM) theory and Chern-Simons (CS) theory with higher-order derivatives. A new form of gauge-ghost proper vertices and new conserved charges at the quantum level are obtained for the YM theory; the quantal Becchi-Rouet-Stora (BRS) conserved charges and conserved angular momentum are also derived for CS theory. The advantage of our canonical formalism is that we do not carry out the integration over the canonical momenta in the phase-space path integral as usually performed.

We consider the sine-Gordon equation under Hamiltonian perturbation with even periodic boundary conditions. We give an analytic expression for homoclinic orbits and produce a useful representation of the gradient of an important integral of motion. We establish the existence of homoclinic tubes based on Mel'nikov analysis and an implicit function theorem argument.

In this work we present the results of a numerical and semiclassical analysis of high-lying states in a Hamiltonian system, whose classical mechanics is of a generic, mixed type, where the energy surface is split into regions of regular and chaotic motion. As predicted by the principle of uniform semiclassical condensation, when the effective tends to zero, each state can be classified as regular or irregular. We were able to semiclassically reproduce individual regular states by the Einstein-Brillouin-Keller torus quantization, for which we devise a new approach, while for the irregular ones we found the semiclassical prediction of their autocorrelation function, in a good agreement with numerics. We also looked at the low-lying states to get a better understanding of the onset of semiclassical behaviour.

We show that a one-dimensional aperiodic Delaunay set of points together with the Fourier transform of its autocorrelation measure (square modulus of its structure factor) at a wavevector k = 2/, can be associated with a generalized Meyer set under some assumptions: (a) that the internal space is toric, / , with a window, assumed finite, equal to the set of affine lattices of period which have a non-empty intersection with and rarefaction laws at infinity, a selection rule based on a congruence mode with respect to ; (b) a scaling exponent function, having values in [0;1], can be uniquely defined on the window from rarefaction laws, which is related to the scaling properties of the intensity function; (c) the projection mappings are adapted to the average lattice of and are not orthogonal. The case of Bragg peaks of the Thue-Morse sequence spectrum is developed explicitly in this context.

We show that the constrained KP hierarchies and their generalizations are natural reductions of the multi-component KP hierarchy and that particular solutions of these hierarchies are obtained in a straightforward way from that of the multi-component KP hierarchy.

On page 1898, definition 1, `C<sup>*</sup>-algebra' should be `*-algebra'.

On page 1905, the line after equation (7.10), `norm' should be `pseudo-norm'.