Table of contents

It is shown that the q-Laguerre and Wall (or little q-Laguerre) polynomials are interrelated by the Fourier - Gauss transform. In the limit when the degree of these polynomials tends to infinity, this integral transform provides the relation between Jackson's second and third q-Bessel functions.

A formal correspondence beetwen the surface magnetization of an Ising quantum chain, perturbed by the paper-folding aperiodic sequence, and the partition function of a classical Ising chain in an inhomogeneous external field is derived. The perturbation is marginal and the critical exponent associated with the surface magnetization is a continuous function of the perturbation amplitude. We obtain this exponent by analysing the classical chain.

It is shown that the standard Gauss - Codazzi system of equations which describe generic surfaces immersed into the three-dimensional Euclidean space is equivalent to the two-dimensional Dirac equation (Davey - Stewartson linear problem) accompanied by a specific constraint. It is demonstrated that this system is linearizable via the parametrization of the moving trihedral by the Euler angles. Some special classes of surfaces are also considered.

In this paper we introduce a three replica potential, useful for examining the structure of metastable states above the static transition temperature, in the spherical p-spin model. Studying the minima of the potential we are able to find the distance between the nearest equilibrium and local equilibrium states, thus obtaining information on the dynamics of the system. Furthermore, the analysis of the potential at the dynamical transition temperature suggests that equilibrium states are not randomly distributed in the phase space.

We study zero temperature properties of a system of two coupled quantum spin chains subject to fields explicitly breaking time reversal symmetry and parity. A suitable choice of the strength of these fields gives a model soluble by Bethe ansatz methods which allows us to determine the complete magnetic phase diagram of the system and the asymptotics of correlation functions from the finite size spectrum. The chiral properties of the system for both the integrable and the nonintegrable case are studied using numerical techniques.

We consider the quasispecies description of a population evolving in both the `master sequence' landscape (where a single sequence is evolutionarily preferred over all others) and the REM landscape (where the fitness of different sequences is an independent, identically distributed, random variable). We show that, in both cases, the error threshold is analogous to a first-order thermodynamical transition, where the overlap between the average genotype and the optimal one drops discontinuously to zero.

We study the dynamics of the SK model modified by a small non-Hamiltonian perturbation. We study aging, and we find that on the timescales investigated by our numerical simulations it survives a small perturbation (and is destroyed by a large one). If we assume that we are observing a transient behaviour, the scaling of correlation times versus the asymmetry strength is not compatible with the one expected for the spherical model. We discuss the slow power law decay of observable quantities to equilibrium, and we show that for small perturbations power-like decay is preserved. We also discuss the asymptotically large time region on small lattices.

We study the one-dimensional partially asymmetric simple exclusion process (ASEP) with open boundaries, that describes a system of hard-core particles hopping stochastically on a chain coupled to reservoirs at both ends. Derrida and coworkers showed in 1993 that the stationary probability distribution of this model can be represented as a trace on a quadratic algebra, closely related to the deformed oscillator-algebra. We construct all finite-dimensional irreducible representations of this algebra. This enables us to compute the stationary bulk density as well as all correlation lengths for the ASEP on a set of special curves of the phase diagram.

Dynamic relaxation of the XY-model quenched from a high temperature state to the critical temperature or below is investigated with Monte Carlo methods. When a non-zero initial magnetization is given, in the short-time regime of the dynamic evolution the critical initial increase of the magnetization is observed. The dynamic exponent is directly determined. The results show that the exponent varies with respect to the temperature. Furthermore, it is demonstrated that this initial increase of the magnetization is universal, i.e. independent of the microscopic details of the initial configurations and the algorithms.

We derive uniform approximations for contributions to Gutzwiller's periodic-orbit sum for the spectral density which are valid close to bifurcations of periodic orbits in systems with mixed phase space. There, orbits lie close together and give collective contributions, while the individual contributions of Gutzwiller's type would diverge at the bifurcation. New results for the tangent, the period-doubling and the period-tripling bifurcation are given. They are obtained by going beyond the local approximation and including higher-order terms in the normal form of the action. The uniform approximations obtained are tested on the kicked top and are found to be in excellent agreement with exact quantum results.

Semiclassical approximations often involve the use of stationary phase approximations. This method can be applied when is small in comparison with relevant actions or action differences in the corresponding classical system. In many situations, however, action differences can be arbitrarily small and then uniform approximations are more appropriate. In this paper we examine different uniform approximations for describing the spectra of integrable systems and systems with mixed phase space. This is done on the example of two billiard systems, an elliptical billiard and a deformation of it, an oval billiard. We derive a trace formula for the ellipse which involves a uniform approximation for the Maslov phases near the separatrix, and a uniform approximation for tori of periodic orbits close to a bifurcation. We then examine how the trace formula is modified when the ellipse is deformed into an oval. This involves uniform approximations for the break-up of tori and uniform approximations for bifurcations of periodic orbits. Relations between different uniform approximations are discussed.

An exact solution of a two-dimensional RSOS model of wetting at a corrugated (periodic) wall is found using transfer matrix techniques. In contrast to mean-field analysis of the same problem, the wetting transition remains second order and occurs at a lower temperature than that of the planar system. A comparison with numerical studies and other analytical approaches is made.

We consider uniform star polymers on a variety of lattices, with an additional contact interaction between pairs of vertices which are unit distance apart and are not joined by an edge of the star. We present some rigorous results and other evidence which indicate that these systems have the same limiting free energy as self-interacting self-avoiding walks. We discuss the extension of these results to other homeomorphism types and to systems with an additional surface interaction term.

The intermediate expansion technique is developed for the overlap coefficients of different biorthogonal coupled states and coupling coefficients of the quantum algebra and group SU(3) with repeating irreducible representations and the role of separate basic and classical hypergeometric functions is demonstrated. Some expansion coefficients such as new prime overlap functions (equivalent to the bilinear combinations of the different boundary isofactors) are expressed in terms of the balanced (Saalschützian) basic or classical hypergeometric series. These new overlap functions, their analytical inversion, some known triangular expansion matrices, transition matrices related to definite Racah coefficients, the well-poised or series and compositions of q-factorial series resembling the well-poised and series or equivalent to and series provide themselves as generators for mutual expansion of different non-orthonormal systems of and SU(3) isofactors, which may be orthonormalized to the paracanonical or (with a definite caution) canonical version.

We discuss the Lie symmetry approach to homogeneous, linear, ordinary differential equations in an attempt to connect it with the algebraic theory of such equations. In particular, we pay attention to the fields of functions over which the symmetry vector fields are defined and, by defining a noncharacteristic Lie subalgebra of the symmetry algebra, are able to establish a general description of all continuous symmetries. We use this description to rederive a classical result on differential extensions for second-order equations.

Boundary equations for the relativistic string with masses at ends are formulated in terms of geometrical invariants of world trajectories of masses at the string ends. In the three-dimensional Minkowski space , there are two invariants of that sort, the curvature K and torsion . Curvatures of trajectories of the string massive ends are always constant, , whereas torsions are the functions of and obey a system of differential equations of second order with deviating arguments. For periodic torsions , where l is the string length in the plane of parameters and , these equations result in constant of motion.

For the wave equations of optics and acoustics of isotropic media the infinite families of three-dimensional plane wave Cauchy operators are found by a direct tensor method. These families form involutive Lie groups. Their generators N can be found by taking square roots from the unit tensors of the wavefront subspaces with outer normal n. In optics the basic structural elements of N are complex involutive operators (reflection isometries) described by pairs of complex vectors S and C, which satisfy the metric condition , and also by a pair of projective operators of the two-dimensional space of a plane orthogonal to n. In the acoustics of isotropic media, in view of the inequality of the longitudinal and transverse wave velocities, the generators N are represented as a linear combination of an involutive operator and a diad . It is shown that the projection of the average energy flux of the wave is conserved in the general case . The families of vectors , , , , , , being a part of N, are indicated. For these families the global operators acting on initial-field vectors give states described by the right-hand and left-hand elliptical helices. The wave normal n characterizes the direction of the angular momentum of the field and for the case turns out to be equivalent to the Darboux vector known in geometry.

The h-deformation of functions on the Grassmann matrix group Gr(2) is presented via a contraction of . As an interesting point, we have seen that, in the case of the h-deformation, both R-matrices of and are the same.

We analyse the way in which duality constrains the exact beta function and correlation length in single-coupling spin systems. We propose a consistency condition which shows very concisely the relation between self-dual points and phase transitions, and implies that the correlation length must be duality invariant. These ideas are then tested on the two-dimensional Ising model, and used towards finding the exact beta function of the q-state Potts model. Finally, a generic procedure is given for identifying a duality symmetry in other single-coupling models with a continuous phase transition.

A recently proposed renormalization scheme can be used to deal with nonrelativistic potential scattering exhibiting ultraviolet divergence in momentum space. A numerical application of this scheme is made in the case of potential scattering with divergence for small r, common in molecular and nuclear physics, by using cut-offs in momentum and configuration spaces. The cut-off is finally removed in terms of a physical observable and model-independent result is obtained at low energies. The expected variation of the off-shell behaviour of the t-matrix arising from the renormalization scheme is also discussed.

We present three different asymptotic studies of the second Painlevé equation involving , or unbounded initial data. We show how the direct method, which is in the spirit of Boutroux, can be naturally applied to each of the three cases.

Given a one-dimensional Sturm - Liouville Schrödinger problem with rational polynomial potential, we can generate the continuous wavelet transform (CWT) for its discrete states, thereby permitting the systematic multiscale reconstruction of the corresponding bound-state wavefunction. A key component in this is the use of properly dilated (a) and translated (b) moments, , which readily transform the configuration space Hamiltonian into a finite set of dynamically coupled, linear, first-order differential equations in the dilation-related variable, :

The infinite scale problem is readily solved through moment quantization methods and used to generate the moments at all scales. We demonstrate the essentials through the rational fraction potential, , and the Coulomb potential.

Using a Hilbert space formalism we present axiomatic models of both a current-fed thick superconducting ring and a dc SQUID (superconducting quantum interference device) as quantum systems possessing superselection rules. A method of quantization by parts is introduced to establish a quantum theory of a system having a circuit configuration. This involves separate quantization of parts of a circuit: the whole system is then recovered by adding these separately quantized parts together. Our models make clear the difference between standard quantum interference and the interference effects exhibited by SQUIDs. They lead us to question a commonly accepted definition of a classical system, and also clarify the properties required of measuring apparatus in the quantum and classical realms.

We study permutation-type solutions to n-simplex equations, that is, solutions whose matrix form can be written as with some matrix A and vector B, both over . With this ansatz the equations of the n-simplex equation reduce to a matrix equation over . We have completely analysed the 2-, 3- and 4-simplex equations in the generic D case. The solutions show interesting patterns that seem to continue to still higher simplex equations.

The prototype quantum master equation is proposed for modelling molecular rotational relaxation caused by isotropic perturbation. Using the detailed balance relation and dissipative properties of the master equation, we can considerably diminish the number of parameters specifying the model. This allows one to evaluate the Heisenberg operator of molecular angular momentum.

Inspired by a recently proposed procedure by Leonhardt and Raymer for wavepacket reconstruction, we calculate the irregular wavefunctions for the bound states of the Coulomb potential. We select the irregular solutions which have the simplest semiclassical limit.

A suitable operator for the time-of-arrival at a detector is defined for the free relativistic particle in (3 + 1) dimensions. For each detector position there exists a subspace of detected states in the Hilbert space of solutions to the Klein - Gordon equation. Orthogonality and completeness of the eigenfunctions of the time-of-arrival operator apply inside this subspace, opening up a standard probabilistic interpretation.

Usually, an integrable nonlinear partial differential equation can be transformed to its conformal invariant form (Schwartz form). Using the conformal invariance of the integrable models, we can obtain many interesting results. In this paper, we will focus mainly in obtaining new symmetries and new integrable models. Starting from the conformal invariance of an integrable model, one can obtain infinitely many non-local symmetries. Many types of (1 + 1)- and (2 + 1)-dimensional new sine - Gordon (or sinh - Gordon) extensions are obtained from the conformal flow equations of the Koerteweg - de Vries type equations. Many other kinds of integrable models can be obtained from the conformal constraints of the known integrable models.

The zero sets of KdV -functions are characterized in terms of the stratification of the infinite Grassmannian. It is shown that these sets are related to integrable hierarchies arising from Schrödinger equations with energy-dependent potentials.

We investigate the Hamiltonian nature of two Miura maps between the modified KP and KP hierarchies. We show that they are canonical, in the sense that the bi-Hamiltonian structure of the modified KP hierarchy is mapped to the bi-Hamiltonian structure of the KP hierarchy.

We study the bound states of a Kronig Penney potential for a nonlinear one-dimensional Schrödinger equation. This potential consists of a large, but not necessarily infinite, number of equidistant -function wells. We show that the ground state can be highly degenerate. Under certain conditions furthermore, even the bound state that would normally be the highest can have almost the same energy as the ground state. This holds for other simple periodic potentials as well.

We give the decomposition of the Kronecker products and the symmetrized Kronecker squares of all the fundamental representations of the harmonic series of unitary irreducible representations of U(p,q). The results for U(2,2) are relevant to two-electron hydrogenic-like atoms.

Uniqueness up to isomorphism, of the Moyal product and bracket of functions on as associative and Lie deformations of the ordinary product and Poisson bracket, is known to follow under additional hypotheses. Using an integral formalism we show here that this result holds without these hypotheses.

The growth of Bargmann functions is intimately connected to the density of the zeros of these functions and to the completeness of sequences of coherent states. Using these ideas we find the least density that a sequence of coherent states must have in order to be overcomplete within the space of Bargmann functions of an order not exceeding (and of a type not exceeding if of order ). These results generalize known results on the completeness of von Neumann lattices. The practical significance of this formalism in the context of quantum optics is also discussed.

This paper deals with the asymptotic behaviour of solutions for the Benjamin - Bona - Mahony equation. We first show the existence of the global weak attractor for this equation in . And then by an idea of Ball, we prove that the global weak attractor is actually the global strong attractor. The finite-dimensionality of the global attractor is also established.

In this paper, the establishment of the dressing transformations in the sinh - Gordon model is reconsidered. By carefully analysing the infinitesimal structures of dressing transformations, we improve the algebraic method for solving the dressing problem in the system and then lay the dressing transformation method on a firm basis. The modified dressing transformation method, which no longer contains any deductive jumps, turns out to become a powerful Hamiltonian approach to finding N-soliton solutions of the integrable systems.