Table of contents

Geometrical disorder is present in many physical situations giving rise to eigenvalue problems. The simplest case of diffusion on a random lattice with fluctuating site connectivities is studied analytically and by exact numerical diagonalizations. Localization of eigenmodes is shown to be induced by geometrical defects, that is sites with abnormally low or large connectivities. We expose a `single defect approximation' (SDA) scheme founded on this mechanism that provides an accurate quantitative description of both extended and localized regions of the spectrum. We then present a systematic diagrammatic expansion allowing to use SDA for finite-dimensional problems, e.g. to determine the localized harmonic modes of amorphous media.

The approach to the dynamics of a charged particle in the Born-Infeld nonlinear electrodynamics developed in Chruscinski (1998 Phys. Lett. A 240 8) is generalized to include Born-Infeld dyons. Both Hamiltonian and Lagrangian are constructed. Some similarities with the BPS mass formula and topological field theory are discussed.

The effects of the surface exchange anisotropy on the ordering of ferromagnetic films are studied for the exactly solvable classical spin-vector model withD components. For small surface anisotropy ´_s<<1 (defined relative to the exchange interaction), the shift of T_c in a film consisting of N<<1 layers behaves as T_c^bulk-T_c(N)(1/N)ln(1/´_s) in three dimensions. The finite-size scaling limit T_c^bulk-T_c(N)1/(´^1/2N²), which is realized for the model with a bulk anisotropy ´<<1 in the range N´^1/21, never appears for the model with the pure surface anisotropy. Here for Nexp(-1/´_s)1 in three dimensions, film orders at a temperature above T_c^bulk (the surface phase transition). In the semi-infinite geometry, the surface phase transition occurs for whatever small values of ´_s (i.e., the special phase transition corresponds to T_c^bulk) in dimensions three and lower.

Motivated by the problem of N-coupled Hubbard chains, we investigate a generalization of a recent model containing two species of one-dimensional fermions interacting via a gauge field that depends on the positions of all the particles of the other species. The exact many-body ground state of the model can be easily obtained through a unitary transformation of the model. The correlation functions are Luttinger like - i.e., they decay through power laws with non-integer exponents. Through the interaction-dependent two-particle correlation functions, we identify the relevant perturbations and hence, possible instabilities. Interestingly, for N>2 bands, beyond a critical strength of the interaction, the dominant incipient instability changes.

A composite polygon is composed of a lattice polygon in the square lattice, which contains in its interior an internal structure, which may also be a lattice polygon, or a lattice tree or a lattice animal, or a lattice disc (or a collection of these). The properties of composite polygons are considered in this manuscript. In particular, I shall consider the growth constants and generating functions of these models, as well as the statistical mechanics of interacting models of composite polygons. It is shown that there is an adsorption transition of the internal structure on the containing polygon, and a transition which corresponds to the inflation of the containing polygon (by the internal structure).

We compute the correlation functions of the eigenvalues in the Gaussian unitary ensemble using the fermionic replica method. We show that non-trivial saddle points, which break replica symmetry, must be included in the calculation in order to correctly reproduce the exact asymptotic behaviour of the correlation functions at a large distance.

We study a model of deposition, diffusion and aggregation of particles forming one-dimensional islands in a square lattice. The dependence of the island density exponent on the anisotropy diffusion parameter A is analysed. It is found that continuously decreases from = (1/3) to = (1/4) when A increases from A = 1 to infinity. This nonuniversal behaviour is a direct consequence of the finite island size and, when A>1, the fast diffusion direction is perpendicular to the growth direction of the islands. For infinite anisotropic diffusion, A = , the anomalous result = 0 is obtained.

We consider high-temperature expansions for the free energy of zero-field Ising models on planar quasiperiodic graphs. For the Penrose and the octagonal Ammann-Beenker tiling, we compute the expansion coefficients up to 18th order. As a by-product, we obtain exact vertex-averaged numbers of self-avoiding polygons on these quasiperiodic graphs. In addition, we analyse periodic approximants by computing the partition function via the Kac-Ward determinant. It turns out that the series expansions alone do not yield reliable estimates of the critical exponents. This is due to the limitation on the order of the series caused by the number of graphs that have to be taken into account, and, more seriously, to rather strong fluctuations in the behaviour of the coefficients. Nevertheless, our results are compatible with the commonly accepted conjecture that the models under consideration belong to the same universality class as those on periodic two-dimensional lattices.

Properties of nonlinear waves in one of the nonlinear partial differential equations are investigated. The kinks with the solitonic, but unusual from the classic viewpoint, behaviour are found. The equation itself does not possess the Painlevé property.

Using the correlation function of the chiral vertex operators of the Coulomb gas model, we find the Laughlin wavefunctions of the quantum Hall effect, with filling factor = 1/m, on Riemann sufaces with a Poincarémetric. The same is done for quasihole wavefunctions. We also discuss their plasma analogy.

A new type of duality symmetry is introduced in the momentum space, by Fourier transforming a system of N-form abelian free fields. Exploiting the complex nature of the mode amplitudes, it is shown that the corresponding duality group is both Z₂ and SO(2) for all even dimensions. The connection with the conventional duality symmetry where the group is Z₂ (SO(2)) for D = 4k+2(4k) spacetime dimensions is discussed in detail.

We compute the influence of an external magnetic field on the Casimir energy of a massive charged scalar field confined between two parallel infinite plates. For this case the obtained result shows that the magnetic field inhibits the Casimir effect.

We consider the photon field between an unusual configuration of infinite parallel plates, namely: a perfectly conducting plate ( ) and an infinitely permeable one (µ ). After quantizing the vector potential in the Coulomb gauge, we obtain explicit expressions for the vacuum expectation values of field operators of the form E_iE_j0 and B_iB_j0. These field correlators allow us to re-obtain the Casimir effect for this set-up and to discuss the light velocity shift caused by the presence of plates (Scharnhorst effect: Scharnhorst (1990 Phys. Lett. B 236 354), Barton (1990 Phys. Lett. B 237 559), Barton and Scharnhorst (1993 J. Phys. A: Math. Gen.26 2037)) for both scalar and spinor QED.

We consider a quantum particle in a waveguide which consists of an infinite straight Dirichlet strip divided by a thin semitransparent barrier on a line parallel to the walls which is modelled by a potential. We show that if the coupling strength of the latter is modified locally, i.e., it reaches the same asymptotic value in both directions along the line, there is always a bound state below the bottom of the essential spectrum provided the effective coupling function is attractive in the mean. The eigenvalues and eigenfunctions, as well as the scattering matrix for energies above the threshold, are found numerically by the mode-matching technique. In particular, we discuss the rate at which the ground state energy emerges from the continuum and properties of the nodal lines. Finally, we investigate a system with a modified geometry: an infinite cylindrical surface threaded by a homogeneous magnetic field parallel to the cylinder axis. The motion on the cylinder is again constrained by a semitransparent barrier imposed on a `seam' parallel to the axis.

The family of quaternionic quasi-unitary (or quaternionic unitary Cayley-Klein algebras) is described in a unified setting. This family includes the simple algebras sp(N+1) and sp(p,q) in the Cartan series C_N+1, as well as many non-semisimple real Lie algebras which can be obtained from these simple algebras by particular contractions. The algebras in this family are realized here in relation with the groups of isometries of quaternionic Hermitian spaces of constant holomorphic curvature. This common framework allows one to perform the study of many properties for all these Lie algebras simultaneously. In this paper the central extensions for all quasi-simple Lie algebras of the quaternionic unitary Cayley-Klein family are shown to be trivial no matter their dimension.

Time delays can occur naturally or as transport lags in many physico-chemical as well as biological systems. Incorporating them into a lumped parameter system results in a system of first-order ordinary delay-differential equations (DDEs). In this paper, we develop two-parameter periodic solutions near a Hopf point in such systems using the general reductive perturbation theory and apply the results to a nonisothermal chemical reactor with delayed feedback. The paper suggests that the two-parameter result can be generalized to multiple time delays and other parameters. Results of this work can be useful in constructing plane wave solutions, rotating waves, phase singularity and other interesting phenomena for temporal kinetic systems with time delays.

It is shown that there may be more abundant solitary wave structures of the nonlinear coupled scalar field than those of single scalar fields. In this paper, starting from a known simple example which is used in particle physics and condensed matter physics, we obtained various exact solitary wave and conoidal wave solutions by solving ⁴, ³, +³, ³+⁴, ⁶, ⁵ and models. Generally, from an arbitrary given single scalar field we may obtain a subset of solutions which are also solutions of the nonlinear coupled scalar fields.

The properties of the set of extended Jordanian twists are studied. It is shown that the boundaries of contain twists whose characteristics differ considerably from those of internal points. The extension multipliers of these `peripheric' twists are factorizable. This leads to simplifications in the twisted algebra relations and helps to find the explicit form for coproducts. The peripheric twisted algebra U(sl(4)) is obtained to illustrate the construction. It is shown that the corresponding deformation U_P(sl(4)) cannot be connected with the Drinfeld-Jimbo one by a smooth limit procedure. All the carrier algebras for the extended and the peripheric extended twists are proved to be Frobenius.

We study the solutions of a family of discrete Painlevé equations. The equations that we examine are given as a system of two first-order non-autonomous mappings. The solutions we are interested in are the ones obtained whenever the Painlevé equation can be reduced to a discrete Riccati equation, which can be linearized through a Cole-Hopf transformation. The special solutions thus obtained involve generalizations or reductions of the hypergeometric (and q-hypergeometric) function.

In the quantization scheme which weakens the hermiticity of a Hamiltonian to its mere T invariance the superposition V(x) = x²+Ze²/x of the harmonic and Coulomb potentials is defined at the purely imaginary effective charges (Ze² = if) and regularized by a purely imaginary shift of x. This model is quasi-exactly solvable: We show that at each excited, (N + 1)th harmonic-oscillator energy E = 2N+3 there exists not only the well known harmonic oscillator bound state (at the vanishing charge f = 0) but also a normalizable (N + 1)-plet of the further elementary Sturmian eigenstates _{n}(x) at eigencharges f = f_{n}<0, n = 0,1, ... ,N. Beyond the smallest multiplicities N we recommend perturbative methods for their construction.