Table of contents

We extend the Yangian double to the super (or graded) case and give its Drinfel'd generators a realization by Gauss decomposition.

The topological pressure is obtained as the leading zero of a dynamical zeta function. We consider the problem of computing this zero when it is close to a singularity. In particular we study a family of intermittent maps, which we argue exhibit a branch point singularity in its zeta functions. The convergence of the cycle expansion close to this point is extremely slow. To deal with this problem we consider a resummation of the cycle expansion. The idea is quite similar to that of Padé approximants, but the ansatz is a generalized series expansion around the branch point rather than a rational function. The improvement on convergence of the leading zero is considerable. We also briefly discuss the relation between correlation decay and the nature of the branch point.

We examine the density - density correlation function in a model recently proposed to study the effect of entropy barriers in glassy dynamics. We find that the relaxation proceeds in two steps with a fast beta process followed by alpha relaxation. The results are physically interpreted in the context of an adiabatic approximation which allows one to separate the two processes and define an effective temperature in the off-equilibrium dynamics of the model. We investigate the behaviour of the response function associated with the density and find violations of the fluctuation dissipation theorem.

We consider diagonal disordered one-dimensional Anderson models with an underlying periodicity. We assume the simplest periodicity, i.e. we have essentially two lattices, one that is composed of the random potentials and the other of non-random potentials. Due to the periodicity, special resonance energies appear, which are related to the lattice constant of the non-random lattice. Further on two different types of behaviour are observed at the resonance energies. When a random site is surrounded by non-random sites, this model exhibits extended states at the resonance energies, whereas otherwise all states are localized with, however, an increase of the localization length at these resonance energies. We study these resonance energies and evaluate the localization length and the density of states around these energies.

From the spectral representation of the transition probability of birth-and-death processes, Karlin and McGregor show that the transition probability for the infinite server Markovian queue is in the form of a diagonal sum involving a product of Charlier polynomials. By using Meixner's bilinear generating formula for the Charlier polynomials and the Markov property, a multiple generating for the Charlier polynomials is deduced from the Chapman - Kolmogorov equation. The resulting formula possesses the same genre of a multiple generating function for the generalized Laguerre polynomials discussed by Messina and Paladimo, the explicit solution of which is recently given by the present author.

We introduce a lattice model, in which frustration plays a crucial role, to describe relaxation properties of granular media. We show Monte Carlo results for compaction in the presence of vibrations and gravity, which compare well with experimental data.

The storage capacity, that is the number of patterns which can be stored per weight, is calculated for the fully-connected committee machine with real couplings and K hidden units from the vanishing of the entropy of the internal representations, and it is found to diverge as .

Elementary excitations in the one-dimensional Hubbard model with boundaries are discussed at the half-filling and without external magnetic fields. The energy of the present model is evaluated in the low-lying excited state, where there exist quasiparticles corresponding to elementary excitations in the charge and the spin sectors. The boundary scattering matrix of the quasiparticles is evaluated.

In this note we consider maps which are defined on continuous space whose large time behaviour displays a strange attractor. We are interested in the properties of the discrete maps that are obtained from these continuous ones by discretizing the space. Such systems behave as disordered dynamical systems. The strange attractor breaks down in many (sometimes one) periodic attractors. We study here the statistical properties of such attractors. Generalizing previous conjectures we propose that the distribution of the attraction basins' sizes is the same as in the random map problem. This result is shown to be in good agreement with numerical experiments.

We study the behaviour of the Binder cumulant related to long-distance correlation functions of the discrete Gaussian model of disordered substrate crystalline surfaces. We exhibit numerical evidence that the non-Gaussian behaviour in the low-T region persists on large-length scales, in agreement with the broken phase being super-rough.

We investigate the Q-state two-dimensional Potts model. The anisotropic interfacial tension is related by duality to the anisotropic correlation length. For Q > 4 we calculate exactly the anisotropic correlation length at the first-order transition point. From the calculated anisotropic correlation length, the equilibrium crystal shape (ECS) is derived via the Wulff construction. The ECS is expressed by means of a simple algebraic curve. Regarding Q as a temperature scale, we show that the Potts model has the same ECS as the eight-vertex model. We discuss a connection between the ECS and the q-deformed pseudo-Euclidian algebra.

We study the generalization ability of a simple perceptron which learns unlearnable rules. The rules are presented by a teacher perceptron with a non-monotonic transfer function. The student is trained in the on-line mode. The asymptotic behaviour of the generalization error is estimated under various conditions. Several learning strategies are proposed and improved to obtain the theoretical lower bound of the generalization error.

The Bethe ansatz equation is solved to obtain analytically the leading finite-size correction of the spectra of the asymmetric XXZ chain and the accompanying isotropic 6-vertex model near the antiferromagnetic phase boundary at zero vertical field. The energy gaps scale with size N as and their amplitudes are obtained in terms of level-dependent scaling functions. Exactly on the phase boundary, the amplitudes are proportional to a sum of square-root of integers and an anomaly term. By summing over all low-lying levels, the partition functions are obtained explicitly. A similar analysis is performed also at the phase boundary of zero horizontal field in which case the energy gaps scale as . The partition functions for this case are found to be that of a non-relativistic free fermion system. From the symmetry of the lattice model under rotation, several identities between the partition functions are found. The scaling at zero vertical field is interpreted as a feature arising from viewing the Pokrovsky - Talapov transition with the space and time coordinates interchanged.

We investigate the dynamics of a three-state stochastic lattice gas, consisting of holes and two oppositely `charged' species of particles, under the influence of an `electric' field, at zero total charge. Interacting only through an excluded volume constraint, particles can hop to nearest-neighbour empty sites. Using a combination of Langevin equations and Monte Carlo simulations, we study the probability distributions of the steady-state structure factors in the disordered phase where homogeneous configurations are stable against small harmonic perturbations. Simulation and theoretical results are in excellent agreement, both showing universal asymmetric exponential distributions.

An analytical method to obtain the steady-state nucleation flux of binary liquid droplets is developed. The method is based on the solution of a Fokker - Planck equation for the concentration of clusters where both number (size) and cluster-energy fluctuations are included. The Fokker - Planck equation is solved in the vicinity of the saddle point. The kinetic prefactor is found to depend on the product of the three eigenvalues of a matrix that describes fluctuations about the critical cluster. The explicit, analytical expression for the total steady-state nucleation rate is applied to the water - ethanol binary system. The model predicts a nucleation rate that is slightly higher than the classical nucleation rate.

Guided by an approach used in the study of membranes we construct the partition function for a discrete Gaussian chain model of polymers. The Hamiltonian is , where describes the chain connectivity, is the inertia tensor, which gives the shape and size of the polymer chain, and tr denotes the trace. Here we consider only the size, , which is obtained from the partition function by differentiating with respect to the conjugate variable . The partition function is expressed in terms of the hypergeometric function , an orthogonal polynomial, where n is the number of monomers and is the ratio of and the coefficient in . The limit corresponds to the random coil, while the limit describes a compact object. We also study the excluded volume problem by discretizing Edwards' continuous self-avoidance term. We obtain dimensionally regularized expressions for the radius of gyration and the end-to-end distance. The short distance cut-off dependence of the continuous model is reproduced.

We calculate the spin - spin correlation function for a (1 + 1)-dimensional free-fermion model displaying crossover from the two-dimensional Ising to Pokrovsky - Talapov critical behaviour. In the Ising limit our results reduce to the well known exact representations reported by Wu et al. Correspondence between the obtained correlation function and an exactly solvable classical Hamiltonian system is established.

Monte Carlo simulations on the SK model have been done to investigate aging processes after a rapid quench from to the spin-glass phase. Taking care of time ranges of simulation, we examine time evolutions of energy of the system, Parisi's overlap distribution function, auto-correlation and clones-correlation functions, distribution functions of the two correlations, and magnetization induced by the field applied after a certain waiting time. The data simulated exhibit a rich variety of aging phenomena. Most of them can be interpreted in a unified way, though qualitatively, by the scenario of growth of quasi-equilibrium domains which we have recently introduced. The results are consistent qualitatively with asymptotic behaviours of some of the basic assumptions and their results in recent analytical theory on the same SK model, so long as the limiting procedures in finite systems are taken properly. Also they suggest that a basin of attraction of one dominant pure state spans almost an entire phase space of the system with a common time-reversal symmetry.

We analyse the Schrödinger wave equation of a two-level or spinorial Hamiltonian, from a classical point of view. An iterative scheme, the coupled mode semiclassical formalism, is proposed, allowing us to deal with the nonadiabatic transfer. As the WKB expansion, it allows the one-dimensional Schrödinger equation to be integrated by successive quadratures.

Finally, we show that time-dependent information can be drawn from the previous, purely stationary, analysis by extending the notion of group velocity. The proposed formalism is thus coherent with an image of multiple trajectories, conforming more to physical behaviour than a single trajectory.

One may obtain, using operator transformations, algebraic relations between the Fourier transforms of the causal propagators of different exactly solvable potentials. These relations are derived for the shape invariant potentials. Also, potentials related by real transformation functions are shown to have the same spectrum generating algebra with Hermitian generators related by this operator transformation.

There is a four-parameter family of point interactions in one-dimensional quantum mechanics. They represent all possible self-adjoint extensions of the kinetic energy operator. If time-reversal invariance is imposed, the number of parameters is reduced to three. One of these point interactions is the familiar function potential but the other generalized ones do not seem to be widely known. We present a pedestrian approach to this subject and comment on a recent controversy in the literature concerning the so-called interaction. We emphasize that there is little resemblance between the interaction and what its name suggests.

The covariant path integral quantization of the theory of the scalar and spinor fields interacting through the Abelian and non-Abelian Chern - Simons gauge fields in 2 + 1 dimensions is carried out using the De Witt - Fadeev - Popov method. The mathematical ill-definiteness of the path integral of theories with pure Chern - Simons' fields is remedied by the introduction of the Maxwell or Maxwell-type (in the non-Abelian case) terms, which make the resulting theories super-renormalizable and guarantees their gauge-invariant regularization and renormalization. The generating functionals are constructed and shown to be the same as those of quantum electrodynamics (quantum chromodynamics) in 2 + 1 dimensions with the substitution of the Chern - Simons propagator for the photon (gluon) propagator. By constructing the propagator in the general case, the existence of two limits; pure Chern - Simons and quantum electrodynamics (quantum chromodynamics) after renormalization is demonstrated.

The Batalin - Fradkin - Vilkovisky method is invoked to quantize the theory of spinor non-Abelian fields interacting via the pure Chern - Simons gauge field and the equivalence of the resulting generating functional to the one given by the De Witt - Fadeev - Popov method is demonstrated.

The S-matrix operator is constructed, and starting from this S-matrix operator novel topological unitarity identities are derived that demand the vanishing of the gauge-invariant sum of the imaginary parts of the Feynman diagrams with a given number of intermediate on-shell topological photon lines in each order of perturbation theory. These identities are illustrated by explicit examples.

It is shown how one can get numerical prediction of quantum mechanical particle behaviour without using the Schrödinger equation. The main steps of this development are the non-differentiability hypothesis, the equations of motion entailed by this hypothesis, and the numerical formulation of a simple one-dimensional problem: the particle in a box.

We show that the well known fact concerning a coincidence between the leading-order WKB and exact quantum mechanical results for an energy spectrum of the Morse Hamiltonian is due to the following property of the bound-state wavefunctions in the complex plane. The logarithmic derivatives of the bound-state eigenfunctions of the Morse Hamiltonian are periodic functions with a pure imaginary period. We show that the Morse potential is the only potential having this property in the following class of potentials: .

The representations of the oscillator algebra introduced by Brzezinski et al (Brzezinski T, Egusquiza J L and Macfarlane A J 1993 Phys. Lett. 311B 202) are classified.

The aim of this paper is to investigate large time behaviour, i.e. stability and growth bounds, of the solutions for a class of stochastic Burgers equations. The analysis is based on some robustness analysis involved with an infinite-dimensional stochastic evolution equation. Various sufficient conditions for a stochastic Burgers equation are obtained to ensure its asymptotic properties. Lastly, several examples are given to illustrate our theory.

It is a well known fact that the Dirac and Kemmer - Duffin equations are the Bhabha equations. We use the method based on the de Sitter group SO(1,4) to show that the Rarita - Schwinger and Bargmann - Wigner equations can also be treated as the Bhabha equations with some subsidiary conditions. This demonstrates that the de Sitter group can be considered as a significant auxiliary group which provides a unified approach to the equations of relativistic quantum theory.

The shape of a surface with constant mean curvature (CMC) has been studied in mathematics and physics related to nonlinear integrable theory and harmonic map (-model) theory. In the study a fictitious (linear) Dirac-type operator appears as a tool of the calculus (Konopelchenko B G and Taimanov I A 1996 J. Phys. A: Math. Gen. 29 1261 - 5).

In this paper, I confine the Dirac field defined in to a thin surface embedded in and obtain a proper Dirac operator for the thin surface. Then it completely agrees with the Dirac-type operator used in the calculus of the CMC surface theory. In other words, the mathematical Dirac-type operator is realized by a physical Dirac particle.

A geometrical framework is presented for modelling general systems of mixed first- and second-order ordinary differential equations. In contrast to our earlier work on non-holonomic systems, the first-order equations are not regarded here as a priori given constraints. Two nonlinear (parametrized) connections appear in the present framework in a symmetrical way and they induce a third connection via a suitable fibred product. The space where solution curves of the given differential equations live, singles out a specific projection among the many fibrations in the general picture. A large part of the paper is about the development of intrinsic tools - tensor fields and derivations - for an adapted calculus along . A major issue concerns the extent to which the usual construction of a linear connection associated with second-order equations fails to work in the presence of coupled first-order equations. An application of the ensuing calculus is presented.

We develop the path integral theory for master equations of general Lindblad form (positive semigroups), describing Markovian open quantum systems. First the Hamiltonian path integral expression for the propagator is derived, which exhibits nicely the decoherence of pairs of phase space histories. A very appealing picture arises in the semiclassical limit where the degree of decoherence is expressible in terms of a phase space decoherence distance functional. For the important class of (effective) Hamiltonians quadratic in the momenta, we derive the Lagrangian version of the path integral propagator. We then evaluate the path integral approximately in a stationary phase approximation, leading to a Van Vleck-type propagator valid under semiclassical conditions. We also derive the propagator for the soluble damped harmonic oscillator in closed form from path integrals. Finally, connections to the active field of stochastic pure-state descriptions of open quantum systems are established, here in particular to linear quantum state diffusion.

Bogomolny's transfer operator has been used to find an analytical solution for the semiclassical energy eigenvalues of a simple two-dimensional integrable system. The system studied consists of a particle moving in an isotropic harmonic oscillator potential plus a potential. The classical trajectories are used to construct the transfer matrix, and an expression is derived for the eigenvalues of this matrix as a function of the energy. These eigenvalue curves yield the semiclassical energy eigenvalues for the quantum system, which turn out to be exactly the same as the results obtained by solving the Schrödinger equation. Some insight into this unexpected agreement is provided by considering an exact transfer operator. We show that when this operator is expanded in powers of Planck's constant, the leading term in the expansion is Bogomolny's transfer operator.

Certain types of generalized undeformed and deformed boson algebras which admit a Hopf algebra structure are introduced, together with their Fock-type representations and their corresponding R-matrices. It is also shown that a class of generalized Heisenberg algebras including those underlying physical models such as that of Calogero - Sutherland, is isomorphic with one of the types of boson algebra proposed, and can be formulated as a Hopf algebra.

Plane waves on symmetric spaces (SS) of rank p, , are constructed by realization of the irreducible representations (principal series) of the group SO(p,q) in the space of infinitely differentiable homogeneous vector functions on cones , , with values in the representation space of the stability subgroups SO(p-i,q-i), i = 1,...,p. We define the cones , , corresponding to the SS X related with Cartan involutive automorphism , , where is the metric tensor of the pseudo-Euclidean space . Calculating Harish - Chandra c-functions the orthogonality, completeness conditions and addition theorems for plane waves are derived. The integrable n-body quantum systems related to groups SO(p,q) are considered. The explicit expressions for the Green functions in the case SS X of rank p = 1 and the integral representation in the general case are given.

Plane waves on symmetric spaces (SS) , are constructed. The integrable n-body quantum systems related to SS X are considered.

We introduce a general formalism to obtain localized quantum wavepackets as dynamically controlled systems, in the framework of Nelson stochastic quantization. We show that in general the control is linear, and it amounts to introducing additional time-dependent terms in the potential. In this way one can construct for general systems either coherent packets following classical motion with constant dispersion, or coherent packets following classical motion whose time-dependent dispersion remains bounded for all times. We show that in the operatorial language our scheme amounts to introducing a suitable generalization to arbitrary potentials of the displacement and scaling operators that generate the coherent and squeezed states of the harmonic oscillator.

The method recently proposed by Skála and Cízek for calculating perturbation energies in a strict sense is ambiguous because it is expressed as a ratio of two quantities which are separately divergent. Even though this ratio comes out finite and gives the correct perturbation energies, the calculational process must be regularized to be justified. We examine one possible method of regularization and show that the proposed method gives traditional quantum mechanical results.

In the published article equation (31), page 6333, is missing. This equation should be:

Also, on page 6335, line 6, delete `and the phase shift' and on pages 6325, 6328, 6335 and 6336 replace Pad by Padé.