Table of contents

Factorization chains for the one-dimensional discrete Schrödinger equation (or the three-term recurrence relation for orthogonal polynomials) defining discrete-time Toda and Volterra lattices are considered. Discrete symmetries, arising from a freedom in the intermediate steps of corresponding double spectral transformations, are described. Some consequences for orthogonal polynomials are discussed.

A standard bicovariant differential calculus on the quantum matrix space is considered. Our main result is proving that the -module differential algebra is in fact a -module differential algebra.

Two types of coherent states for the two-parameter deformed multimode oscillator system are investigated. Moreover, two-parameter deformed gl(n) algebra and deformed symmetric states are constructed.

The spectral shape of one-dimensional systems (describing for instance the behaviour of Frenkel excitons) is approached through the exactly solvable model of the XXO Heisenberg quantum spin chain in a transverse magnetic field. Some results for finite size chains concerning 2N-point correlators are presented in detail. In particular, the finite lattice, finite temperature 2-point correlators are explicitly worked out. Moreover, results in closed form are given for 2N-point correlators in the most general situation (finite lattice and thermodynamic limit, finite temperature, finite space and/or time separations). Their relations with frequency moments of the spectral shape are pointed out and the connection with moment expansion through continued fraction representation is given.

We discuss how to characterize the behaviour of a chaotic dynamical system depending on a parameter that varies periodically in time, such that the time scale of the periodic variation of the parameter is much larger than the `internal' time scale. In particular, we study the predictability time, the correlations and the mean responses, by defining a local-in-time version of these quantities. We find that the local quantities strongly depend on the phase of the cycle. In this case, the standard global quantities can give misleading information.

By means of rather general arguments, based on an approach due to Derrida that makes use of samples of finite size, we analyse the effective diffusivity and drift tensors in certain types of random media in which the motion of the particles is controlled by molecular diffusion and a local flow field with known statistical properties.

The power of the Derrida method is that it uses the equilibrium probability distribution, which exists for each finite sample, to compute asymptotic behaviour at large times in the infinite medium. In certain cases, where this equilibrium situation is associated with a vanishing microcurrent, our results demonstrate the equality of the renormalization processes for the effective drift and diffusivity tensors. This establishes, for those cases, a Ward identity previously verified only to two-loop order in perturbation theory in certain models.

The technique can also be applied to media in which the diffusivity exhibits spatial fluctuations. We derive a simple relationship between the effective diffusivity in this case and that for an associated gradient drift problem that provides an interesting constraint on previously conjectured results.

The (001) facets of Si and Ge have a uniaxial structure which changes direction at each mono-atomic step. This type of topology is unusual and new for the theory of surface roughening and reconstruction phase transitions. It can be incorporated into the solid-on-solid model description. The phase diagram includes pre-roughening transitions and disordered flat phases without the need for step - step interactions. The competition between this and the reconstruction in Si(001) can be described by a generalized 4-state clock-step model. This leads to the prediction that Si(001) and Ge(001) undergo a pre-roughening induced simultaneous deconstruction transition.

We investigate a three-dimensional Ising action which corresponds to a class of models defined by Savvidy and Wegner, originally intended as discrete versions of string theories on cubic lattices. These models have vanishing bare surface tension and the couplings are tuned in such a way that the action depends only on the angles of the discrete surface, i.e. on the way the surface is embedded in . Hence the name gonihedric by which they are known. We show that the model displays a rather clear first-order phase transition in the limit where self-avoidance is neglected and the action becomes a plaquette one. This transition persists for small values of the self avoidance coupling, but it turns to second-order when this latter parameter is further increased. These results exclude the use of this type of action as models of gonihedric random surfaces, at least in the limit where self avoidance is neglected.

A trail is a walk on a lattice that may visit a site more than once but a bond at most once. We have carried out transfer-matrix studies of trails on the square lattice and of hybrid walks that interpolate between self-avoiding walks and trails. The results are in agreement with the same universal exponents as self-avoiding walks. However, the finite-size corrections are much larger than for self-avoiding walks. An explanation in terms of an irrelevant variable with scaling index is given.

A recently proposed form of dual theory for three dimensional superconductor is rederived starting from the lattice electrodynamics and studied by renormalization group. The superfluid density below and close to the transition vanishes as an inverse of the correlation length for the disorder field. The corresponding universal amplitude is given by the fixed point value of the dual charge, and it is calculated to the leading order. The continuum dual theory predicts the divergence of the magnetic field penetration depth with the XY exponent, in contradiction to the results obtained from the Ginzburg - Landau theory for the superconducting order-parameter. Possible reasons for this difference are discussed.

We introduce a contact model with evaporation and deposition of particles at rates p and (1-p), respectively, per occupied lattice site; while the deposition probability on empty sites depends on the number of occupied nearest-neighbour sites. At large times t this model has three different phases, separated by two critical points ( and ). Such phases are: (i) The growth phase . Here the mean value of particles per lattice site n and its fluctuations w always increase as time increases. However, two different regimes can be observed, that is and , for ; while just at one has . (ii) The steady-state phase , in which n and w reach finite non trivial (n > 0 and w > 0) values, but both quantities diverge for as . (iii) The inactive (or vacuum) state , for which n=0. At the system exhibits an irreversible phase transition which belongs to the universality class of directed percolation model, so for , and , with . Transitions between phases are continuous, however, the transition at is reversible (irreversible), respectively.

The aim of the present work is to derive the first quantum correction to the fourth virial coefficient for fluids of molecules interacting according to the square-well potential of arbitrary dimensionality d. In this paper, we show that the basic method of Hemmer and Jancovici, which was followed by Gibson, can be extended to cover more general intermolecular potentials. The extension of the formalism is straightforward, but some consideration has to be given to the problem of how to truncate the resulting expansion to get all the quantum corrections to a given order in . The first quantum correction to the fourth virial coefficient is obtained in arbitrary dimensionality (d=1,3).

We consider determining the configurational properties of a neutral polymer in two dimensions (2D) via self-consistent mean-field methods. By suitably scaling the problem we recover the Flory result for polymers under the excluded volume interaction, i.e. , where is the mean scaling length of a polymer which consists of (N+1) monomers. If we let x denote the scaled distance from one end of the polymer to a point in space we find that there exists a point , where the scaled polymer density , decays rapidly to zero. Physically the existence of such a point is expected since the polymer has a finite length. For we find while for we obtain . We discuss the consequence of these results on the validity of the asymptotic methods used.

We study numerically the probability distribution of the Yang - Lee zeroes inside the Griffiths phase for the two-dimensional site diluted Ising model and we check that the shape of this distribution is that predicted in previous analytical works. By studying the finite-size scaling of the averaged smallest zero at the phase transition we extract, for two values of the dilution, the anomalous dimension, , which agrees very well with the previous estimated values.

We study the q-state Potts antiferromagnet with q=3 on the honeycomb lattice. Using an analytic argument together with a Monte Carlo simulation, we conclude that this model is disordered for all . We also calculate the ground state entropy to be and discuss this result.

We show how non-periodic (quasiperiodic) long-range order can arise in one dimension due to long-range, translation-invariant pair interactions with quasiperiodically alternating signs. We discuss the Parisi overlap distribution between the infinitely many pure states.

We study the support (i.e. the set of visited sites) of a t-step random walk on a two-dimensional square lattice in the large t limit. A broad class of global properties, M(t), of the support is considered, including for example the number, S(t), of its sites; the length of its boundary; the number of islands of unvisited sites that it encloses; the number of such islands of given shape, size, and orientation; and the number of occurrences in space of specific local patterns of visited and unvisited sites. On a finite lattice we determine the scaling functions that describe the averages, , on appropriate lattice size-dependent time scales. On an infinite lattice we first observe that the all increase with t as , where k is an M-dependent positive integer. We then consider the class of random processes constituted by the fluctuations around average . We show that, to leading order as t gets large, these fluctuations are all proportional to a single universal random process, , normalized to . For the probability law of tends to that of Varadhan's renormalized local time of self-intersections. An implication is that in the long time limit all are proportional to .

We discuss the symmetry decomposition of the average density of states for the two-dimensional potential and for its three-dimensional generalization . In both problems, the energetically accessible phase space is non-compact due to the existence of infinite channels along the axes. It is known that in two dimensions the phase space volume is infinite in these channels thus yielding non-standard forms for the average density of states. Here we show that the channels also result in the symmetry decomposition having a much stronger effect than in potentials without channels, leading to terms which are essentially leading order. We verify these results numerically and also observe a peculiar numerical effect which we associate with the channels. We additionally show that the next-order corrections are anomalously weak, being at least two powers of smaller than one would expect. In three dimensions, the volume of phase space is finite and the symmetry decomposition follows more closely that for generic potentials - however, there are still non-generic effects related to some of the group elements.

The case of large intercentre distance in the two Coulomb centres problem is studied by solving separated wave equations with the help of a series of confluent hypergeometric functions. By considering the confluence of two singularities in an auxiliary equation with four regular singularities, new relations between the solutions of the quasi-angular equation are found and used to obtain exponentially small terms in the asymptotic expansion for energy eigenvalues. For some electronic states, energy splittings at pseudocrossings are evaluated, and results are compared with those of earlier asymptotic and numerical calculations.

The uniaxial bianisotropic-ferrite medium, which can be fabricated by polymer synthesis techniques, is a generalization of the well-studied chiral medium. It has potential applications in the design of antireflection coating, antenna radomes, and novel microwave components. In the present investigation, based on the concept of spectral eigenwaves, eigenfunction expansion of the Green dyadics in this class of materials is formulated in terms of the cylindrical vector wavefunctions. The formulations are greatly simplified by analytically evaluating the integrals with respect to the spectral longitudinal and radial wavenumbers, respectively. The analysis indicates that the solutions of the source-incorporated Maxwell's equations for a homogeneous uniaxial bianisotropic-ferrite medium are composed of two (or four) eigenwaves travelling with different wavenumbers. Each of these eigenwaves is a superposition of two transverse waves and a longitudinal wave. The Green dyadics of planarly and cylindrically multilayered structures consisting of uniaxial bianisotropic-ferrite media can be straightforwardly obtained by applying the method of scattering superposition and appropriate electromagnetic boundary conditions respectively. The resulting formulations, which can be theoretically verified by comparing their special forms with existing results, provide fundamental basis to analyse the physical phenomena of unbounded and multilayered uniaxial bianisotropic-ferrite media.

We analyse the phase flow evolution of the torque free asymmetric gyrostat motion. The gyrostat consists of a triaxial rigid body and a symmetric rotor spinning around one of the principal axis of inertia of the gyrostat. The problem is converted into a two parametric quadratic Hamiltonian with the phase space on the sphere. As the parameters evolve, the appearance - disappearance of centres and saddle points is originated by a sequence of pitchfork bifurcations. When the gyrostat is axial symmetric, there are motions of the rotor that break the degeneracy through an oyster bifurcation while other motions simply shift the degeneracy along a minor circle.

Recently, having reconsidered the reproducing kernel for gauge-invariant states which involves the projection operator onto the reduced Hilbert space of physical states, John Klauder has shown how the phase space coherent state path integral quantization of constrained systems avoids any gauge-fixing conditions, and leads to a specific measure for the integration over Lagrange multipliers. Here, it is pointed out that independently of the coherent state formulation, this approach is also devoid of any Gribov problems and always provides for an effectively admissible integration over all gauge orbits of gauge-invariant systems. This important aspect of Klauder's reappraisal of the physical reproducing kernel is explicitly confirmed by two simple examples.

An algorithm to generate integrable systems is extended to the super case. Some new examples of superextensions of integrable systems are illustrated. We also generalize the trace identity due to Tu to the super case and use it to establish Hamiltonian structures of superextensions of integrable systems under consideration.

The methods of Lie group analysis of differential equations are generalized so as to provide an infinitesimal formalism for calculating symmetries of difference equations. Several examples are analysed, one of them being a nonlinear difference equation. For the linear equations the symmetry algebra of the discrete equation is found to be isomorphic to that of its continuous limit.

The quantum double construction of a q-deformed boson algebra possessing a Hopf algebra structure is carried out explicitly. The R-matrix thus obtained is compared with the existing literature.

In physical applications of differential geometry, one sometimes wishes to compute the holonomy group of a Riemannian manifold from local data, such as the curvature tensor. In general, this can be a complicated problem, but we show that, in cases of most interest in physics, the holonomy group can be obtained directly from the Lie algebras generated by the curvature tensor.

We study the phenomenon of avoided crossings of eigenvalue curves for boundary value problems related to differential equations of Heun's class. The eigenvalues are given explicitly in asymptotic form taking into account power-type as well as exponentially small terms. It is exhibited that the phenomenon of avoided crossings of eigenvalue curves show a `periodical' structure in the sense that at any integer value of the additional controlling parameter an infinite (in the sense of a large parameter) number of avoided crossings take place simultaneously. Some relations to other phenomena of the asymptotics of exponentially small terms are discussed at the end of the article.

A formula is derived for the Nijenhuis tensor of an endomorphism constructed from contracting a bivector field with a 2-form. Two applications are considered: the first is to the uniqueness aspect of the inverse problem of the calculus of variations, the second is to bi-Hamiltonian systems.

A recent normal-form approximation for dynamical equilibria of one-dimensional Hamiltonian systems is shown to provide a phase-integral (WKB) approximation to solutions of nonlinear differential equations. In the present paper, a restricted class of ordinary differential equations , is considered. The integrability of the truncated normal form allows for expressing the solutions as trigonometric expansions in terms of an `amplitude' and `phase'. The method is applied to a Dirichlet boundary value problem on the interval for n=3 and n=4 where the coefficient functions depend on an additional parameter . As in the constant coefficient case, we obtain approximate expressions for eigenvalues and eigensolutions near the linear limit. The results show that the interpretative and the predictive power of the linear WKB solutions carry through to the nonlinear regime of small-amplitude, wavelike solutions . We further analyse the mechanism by which the `odd-n' nonlinearity in general causes a splitting of the linear eigenvalues. In particular, we discuss the singular threshold behaviour of the doubling mechanism for nonlinearities with n=3. If the coefficient functions become constants, the doubling of eigenvalues corresponding to standing waves of odd numbers of nodes gradually disappears. The method of approximation can be worked out similarly for any `perturbing' polynomial in .

The graded reflection equations are presented to describe quantum integrable lattice fermion open chains on a finite interval with independent boundary conditions on each end. Specifically, the general boundary K supermatrices are determined for the one-dimensional small-polaron open chain. The result is consistent with the system under study admitting Lax pair formulation.

We consider the large-N Sutherland model in the Hamiltonian collective-field approach based on the 1/N expansion. The Bogomol'nyi limit appears and the corresponding solutions are given by static-soliton configurations. They exist only for , i.e. for the negative coupling constant of the Sutherland interaction. We determine their creation energies and show that they are unaffected by higher-order corrections.

The Einstein - Podolsky - Rosen argument on quantum mechanics incompleteness is formulated in terms of elements of reality inferred from joint (as opposed to alternative) measurements, in two examples involving entangled states of three spin- particles. The same states allow us to obtain proofs of the incompatibility between quantum mechanics and elements of reality.

We explore the detailed topological structure of the phase-space of the recently discovered three-dimensional integrable but nonseparable Hamiltonian system with velocity dependent potential. Two three-parameter families of three-dimensional tori which foliate the phase-space are identified. The complete classification, according to their topology, of the level sets corresponding to the critical points of the energy-momentum map, is accomplished. The relationship of the three-dimensional integrable system with important physical systems of current interest, namely, the hydrogen atoms in circularly polarized (CP) fields, in crossed magnetic and electric fields and in crossed magnetic and CP fields, is discussed.

An alternative approach to that described in [1] is developed for analytically inverting a particular type of tridiagonal matrix. The technique is then extended to deal with a general banded matrix in which diagonal elements are identical and are flanked in each row by the same set of quantities.

The first line of equation (7) should be:

The last line in the first paragraph on page 6250, after equation (32), should read: `The expressions ; describe the front surfaces of in the case and '

The published article contained a number of errors as follows.

Page 4169, Introduction, line 6: replace `deformations of' by `given by'.

Page 4170, equation (1.2): close gap between and q₀(x).

Page 4170, equation (1.3 b): the matrix entries of R_n have been displaced - the 1 should be below Y(n) and the -k_n above the 0.

Page 4171, equation (1.4): M_n is a 2 × 2 matrix - insert a comma before each - sign.

Page 4171, third line from bottom: insert a `)' at the end of the line before the exponent 2.

Page 4172, line 7: add the argument (x) for .

Page 4172, section 2, line 6: replace w by w.

Page 4173, equation (2.9): replace w by w.

Page 4174, lines 4, 6, 9, 10 (twice), 15, 17: replace w by w.

Page 4174, second paragraph, line 1: replace S by .

Page 4175, figure captions 2 and 3: replace w by w.

Page 4175, second line from bottom: replace `second step' by ` second step'.

Page 4177, line 2: replace w by w.

Page 4177, line 6: replace `ortho normal' by `ortho normal'.

Page 4177, equation (2.28 b): replace by w.

Page 4177, first line after equation (2.28 d): replace by w.

Page 4177, equations (2.30 a) and (2.30 b): replace by .

Page 4178, first line after equation (3.1): delete `)' after Y_n(x).

Page 4180, equation (4.3): delete superscript (N,r) at dσ_Dirac.

Page 4180, second line from bottom: replace kth by kth.

Page 4182, equation (4.10): the second factor under the square-root should be (x-(2r+1)/2r).