Table of contents

In early work, Lawes and March obtained a differential equation for an arbitrary number of independent harmonically confined fermions in one dimension, and very recently this result has been generalized to apply to three-dimensional (3D) isotropic harmonic confinement. Here, an exact solution of this 3D equation for the fermion particle density ρ(r) is constructed, and the near-diagonal form of the Dirac density matrix is also obtained.

We analyse new data for self-avoiding polygons (SAPs), on the square and triangular lattices, enumerated by both perimeter and area, providing evidence that the scaling function is the logarithm of an Airy function. The results imply universal amplitude combinations for all area moments and suggest that rooted SAPs may satisfy a q-algebraic functional equation.

This paper develops a theory for characterization of DNA sequences based on their measure representation. The measures are shown to be random cascades generated by an infinitely divisible distribution. This probability distribution is uniquely determined by the exponent function in the multifractal theory of random cascades. Curve fitting to a large number of complete genomes of bacteria indicates that the gamma density function provides an excellent fit to the exponent function, and hence to the probability distribution of the complete genomes.

We investigate the transport of a passive tracer in a two-dimensional stratified random medium with flow parallel and perpendicular to the strata. Assuming a Gaussian random flow with a Gaussian correlation function, it is not only possible to derive exact expressions for the temporal behaviour of the dispersion coefficients characterizing the tracer transport, but also to investigate the self-averaging properties of these quantities. We give explicit results for the dispersion coefficient and its mean square fluctuations as a function of time. As it turns out, the sample-to-sample fluctuations which are encoded in the latter quantity remain finite even in the asymptotic limit of infinite times, which implies that the given two-dimensional model is not self-averaging.

We investigate the distribution of the time spent by a random walker to the right of a boundary moving with constant velocity v. For the continuous-time problem (Brownian motion), we provide a simple alternative proof of Newman's recent result (Newman T J 2001 J. Phys. A: Math. Gen.34 L89) using a method developed by Kac. We then discuss the same problem for the case of a random walk in discrete time with an arbitrary distribution of steps, taking advantage of the general set of results of Sparre Andersen. For the binomial random walk we analyse the corrections to the continuum limit on the example of the mean occupation time. The case of Cauchy-distributed steps is also studied.

We extend the quasi-continuum method of approximation to the case of a diatomic lattice. This is illustrated by a lattice in which both small and large atoms interact with first- and second-nearest neighbours. We show that highly accurate quasi-continuum techniques may be generalized to determine the shape of nontopological kinks. Many previous analyses of such systems have found travelling wave solutions only for one particular speed using a second-order continuum theory; we show (i) how wave shapes can be found for arbitrary speeds, and (ii) how solution profiles can be calculated from fourth-order partial differential equations which approximate the lattice. Alongside the theoretical analysis, we also present numerical simulations of the lattice which demonstrate propagation of the predicted waves through the lattice. We show that the particular speed for which solutions have been found in previous studies is a special speed for waves in the lattice, but waves can travel for long periods of time at faster or slower speeds, whilst slowly relaxing to this critical speed.

We provide a statistical characterization of the ionization yield of one-dimensional, periodically driven Rydberg states of atomic hydrogen, in the spirit of Anderson localization theory. We find excellent agreement with predictions for the conductance across an Anderson localized, quasi-one-dimensional, disordered wire, in the semiclassical limit of highly excited atomic initial states. For the moderate atomic excitations typically encountered in state of the art laboratory experiments, finite-size effects induce significant deviations from the solid state picture. However, large-scale fluctuations of the atomic conductance prevail and are robust when averaged over a finite interval of driving field amplitudes, as inevitably done in the experiment.

Resonances of the (Frobenius-Perron) evolution operator for phase-space densities have recently attracted considerable attention, in the context of interrelations between classical and quantum dynamics. We determine these resonances as well as eigenvalues of for Hamiltonian systems with a mixed phase space, by truncating to finite size in a Hilbert space of phase-space functions and then diagonalizing. The corresponding eigenfunctions are localized on unstable manifolds of hyperbolic periodic orbits for resonances and on islands of regular motion for eigenvalues. Using information drawn from the eigenfunctions we reproduce the resonances found by diagonalization through a variant of the cycle expansion of periodic-orbit theory and as rates of correlation decay.

Three-dimensional quadratic gravity, unlike general relativity in (2+1)D, is dynamically nontrivial and has a well behaved nonrelativistic potential. Here we analyse the changes that occur when a topological Chern-Simons term is added to this theory. It is found that the harmless massive scalar mode of the latter gives rise to a troublesome massive spin-0 ghost, while the massive spin-2 ghost is replaced by two massive physical particles both of spin 2. We also found that light deflection does not have the `wrong sign' such as in the framework of three-dimensional quadratic gravity.

The SU_q(2) algebra is extended by introducing additional raising and lowering operators and constructing their coherent states. This new algebra of coherent states and the commutation relations between the extended operators are investigated and a resolution of unity is proposed.

We present a complete derivation of the semiclassical limit of the coherent-state propagator in one dimension, starting from path integrals in phase space. We show that the arbitrariness in the path integral representation, which follows from the overcompleteness of the coherent states, results in many different semiclassical limits. We explicitly derive two possible semiclassical formulae for the propagator, we suggest a third one, and we discuss their relationships. We also derive an initial-value representation for the semiclassical propagator, based on an initial Gaussian wavepacket. It turns out to be related to, but different from, Heller's thawed Gaussian approximation. It is very different from the Herman-Kluk formula, which is not a correct semiclassical limit. We point out errors in two derivations of the latter. Finally we show how the semiclassical coherent-state propagators lead to WKB-type quantization rules and to approximations for the Husimi distributions of stationary states.

We construct a wide class of finite -algebras as truncations of Yangians. These truncations correspond to algebra homomorphisms and allow us to construct the -algebras as exchange algebras, the R-matrix being the Yangian one.

As an application, we classify all irreducible finite-dimensional representations of these -algebras and determine their centre.

A general calculation of Casimir energies - in an arbitrary number of dimensions - for massless quantized fields in spherically symmetric cavities is carried out. All the most common situations, including scalar and spinor fields, the electromagnetic field and various boundary conditions are treated with the uppermost accuracy. The final results are given as analytical, closed expressions in terms of Barnes zeta functions. A direct numerical evaluation of the formulae is then performed, which yields highly accurate numbers of, in principle, arbitrarily good precision.

We consider the spectral problem associated with the evolution Schrödinger equation, (D² + k²)ϕ = uϕ, where u is a matrix-square-valued function, with entries in the Schwartz class defined on the real line. The solution ϕ, called the wavefunction, consists of a function of one real variable, matrix-square-valued with entries in the Schwartz class. This problem has been dealt for symmetric potentials u. We found for the present case that the bound states are localized similarly to the scalar and symmetric cases, but by the zeroes of an analytic matrix-valued function. If we add an extra condition to the potential u, we can determine these states by an analytic scalar function. We do this by generalizing the scalar and symmetric cases but without using the fact that the Wronskian of a pair of wavefunction is constant.

According to Etingof and Varchenko, the classical dynamical Yang-Baxter equation is a guarantee for the consistency of the Poisson bracket on certain Poisson-Lie groupoids. Here it is noticed that Dirac reductions of these Poisson manifolds give rise to a mapping from dynamical r-matrices on a pair ⊂ to those on another pair ⊂, where ⊂⊂ is a chain of Lie algebras for which admits a reductive decomposition as = + . Several known dynamical r-matrices appear naturally in this setting, and its application provides new r-matrices, too. In particular, we exhibit a family of r-matrices for which the dynamical variable lies in the grade zero subalgebra of an extended affine Lie algebra obtained from a twisted loop algebra based on an arbitrary finite-dimensional self-dual Lie algebra.

The Watson integral for the d-dimensional hypercubic lattice

and the associated logarithmic integral

are investigated. In particular, a new method is developed which enables one to calculate the numerical values of {L_d : d = 1,2,...} and {W_d : d = 3,4,...} with extremely high precision. The asymptotic behaviour of L_d and W_d as d→∞ is also determined. Finally, some generalizations of the results are briefly discussed.

Using the Weiss method of truncated singular expansions, we construct an explicit Bäcklund transformation of the Drinfeld-Sokolov-Satsuma-Hirota system into itself. Then we find all the special solutions generated by this transformation from the trivial zero solution of this system.

The inverse spectral transform for the Zakharov-Shabat equation on the semi-line x>0 is reconsidered as a Hilbert problem. The boundary data induce an essential singularity as k→∞ to one of the basic solutions. Then solving the inverse problem means solving a Hilbert problem with particular prescribed behaviour. It is demonstrated that the direct and inverse problems are solved in a consistent way as soon as the spectral transform vanishes as (1/k) at infinity in the whole upper half-plane (where it may possess single poles) and is continuous and bounded on the real k axis. The method is applied to stimulated Raman scattering and sine-Gordon (light cone) for which it is demonstrated that time evolution conserves the properties of the spectral transform.

We derive a semiclassical expression for an energy-smoothed autocorrelation function defined on a group of eigenstates of the Schrödinger equation. The system we consider is an energy-conserved Hamiltonian system possessing time-invariant symmetry. The energy-smoothed autocorrelation function is expressed as a sum of three terms. The first one is analogous to Berry's conjecture, which is a Bessel function of the zeroth order. The second and the third terms are trace formulae made from special trajectories. The second term is found to be direction dependent in the case of spacing averaging, which agrees qualitatively with previous numerical observations in high-lying eigenstates of a chaotic billiard.

It is argued that colloidal fractals are self-similar because all computed fractal dimensions of a given fractal have the same value. However, in a mathematical sense the self-similar character of a set implies not only that all fractal dimensions agree in value but also the existence of a set of similarities that generates the fractal. In this paper we report some results from two-dimensional aggregation experiments at the air-liquid interface and show that, with the appropriate selection for the similarities, the colloidal fractal aggregates can be considered as self-similar in a more mathematically rigorous form.

Two different theoretical formulations of the finite temperature effects have been recently proposed for integrable field theories. In order to decide which of them is the correct one, we perform for a particular model an explicit check of their predictions for the one-point function of the trace of the stress-energy tensor, a quantity which can be independently determined by the thermodynamical Bethe ansatz.

A hierarchy of functions including the elliptic gamma function is introduced. It can be interpreted as an elliptic analogue of the multiple gamma function and its trigonometric limit coincides with a q-analogue of the multiple gamma function. Some properties of the functions are considered.

We introduce suitable coordinate systems for interacting many-body systems with invariant manifolds. These are Cartesian in coordinate and momentum space and chosen such that several components are identically zero for motion on the invariant manifold. In this sense these coordinates are collective. We make a connection to Zickendraht's collective coordinates and present certain configurations of few-body systems where rotations and vibrations decouple from single-particle motion. These configurations do not depend on details of the interaction.