Table of contents

In this letter we prove that the half-string three-vertex in closed string field theory satisfies the general gluing and resmoothing theorem. We also demonstrate how one calculates amplitudes in the half-string approach to closed string field theory, by working out explicitly a few simple three-amplitudes.

We analyse the density of roots of random polynomials where each complex coefficient is constructed of a random modulus and a fixed, deterministic phase. The density of roots is shown to possess a singular component only in the case for which the phases increase linearly with the index of coefficients. This means that, contrary to earlier belief, eigenvectors of a typical quantum chaotic system with some antiunitary symmetry will not display a clustering curve in the stellar representation. Moreover, a class of time-reverse invariant quantum systems is shown, for which spectra display fluctuations characteristic of orthogonal ensemble, while eigenvectors confer to predictions of unitary ensemble.

Consistent implementation of the additivity of physical observables in second quantization uniquely establishes quantum statistics. In this scheme the usual Weyl - Heisenberg algebra induces Maxwell - Boltzmann distribution, while Bose - Einstein statistics are related to su(1,1).

Linear partial differential equations of arbitrary order invariant under the Galilei transformations are described. Symmetry classification of potentials for these equations in two- dimensional space is carried out. High-order nonlinear partial differential equations invariant under the Galilei, extended Galilei and full Galilei algebras are studied.

We propose a three-parametric extension of integrable non-uniform spin lattices in 1D. In our model, the spins are located at the equilibrium positions of particles interacting via the potential , the particles being confined by some external field. The Lax representation, conserved current and special set of eigenvectors are presented in explicit form.

The cell geometry of the six-dimensional bcc lattice is investigated. Via klotz construction two different classes of icosahedrally projected quasiperiodic tilings are defined. For both cases we determine the acceptance domains of tiles and give a detailed description of the geometry of all tiles.

We present some exact results on percolation properties of the Ising model, when the range of the percolating bonds is larger than the nearest neighbours. We show that the phase diagram for next-nearest neighbour percolation can be exactly obtained from the nearest-neighbour case, which implies that the percolation threshold, , is still equal to the Ising critical temperature . In addition, we present Monte Carlo calculations of the finite size behaviour of the correlated resistor network defined on the Ising model. The thermal exponent, t, of the conductivity that follows from it is found to be . We observe no corrections to scaling in its finite size behaviour.

In this paper we carry out quantum Monte Carlo simulations of a quantum particle in a one-dimensional random potential (plus a fixed harmonic potential) at a finite temperature. This is the simplest model of an interface in a disordered medium and may also pertain to an electron in a dirty metal. We compare with previous analytical results, and also derive an expression for the sample-to-sample fluctuations of the mean square displacement from the origin which is a measure of the glassiness of the system. This quantity as well as the mean square displacement of the particle are measured in the simulation. The similarity to the quantum spin glass in a transverse field is noted. The effect of quantum fluctuations on the glassy behaviour is discussed.

We investigate the two-point correlations in the quantum spectrum of the square billiard. This system is unusual in that the degeneracy of the energy levels increases in the semiclassical limit in such a way that the average level separation is not given by the inverse of the mean density of states. Hence, for example, the standard level spacings distribution does not tend to the Poissonian limit expected for integrable systems. In this paper we calculate the leading-order asymptotic form of a degeneracy-weighted two-point correlation function using a combination of probabilistic techniques and classical number theory. The result exhibits number-theoretical fluctuations about a mean which is a sum of two terms: one having the usual (constant) Poissonian form and the second representing a small correction which decays as the inverse of the correlation distance. This is confirmed by numerical computations.

In this paper we calculate the finite-size corrections of an anisotropic integrable spin chain, consisting of spins s = 1 and . The calculations are done in two regions of the phase diagram with respect to the two couplings and . In the case of conformal invariance we obtain the final answer for the ground state and its lowest excitations, which generalizes earlier results.

We extend a previously developed technique for computing spin - spin critical correlators in the two-dimensional (2D) Ising model, to the case of multiple correlations. This enables us to derive Kadanoff - Ceva's formula in a simple and elegant way. We also exploit a doubling procedure in order to evaluate the critical exponent of the polarization operator in the Baxter model. Thus we provide a rigorous proof of the relation between different exponents, in the path-integral framework.

Aggregation phenomena of elementary particles into clusters is one of the most fascinating and challenging problems of statistical physics. Here we adopt a stochastic approach for the modelling of these phenomena. More precisely, we formulate the problem as follows (which we shall refer to as the `cavity' method): given a population of N atoms partitioned into p groups, how does a new atom eventually connect to any of these p groups forming a new partition of N + 1 atoms into a certain number of groups? Depending on this local `logic' of pattern formation, the asymptotic structure of groups (in the thermodynamic limit ) can be quite different; also the group size distributions may vary widely.

We prove that, in constrast to the theories of continuous observation, in the formalism of event enhanced quantum theory the stochastic process generating sample histories of pairs (observed quantum system, observing classical apparatus) is unique. This result gives a rigorous basis to the previous heuristic argument of Blanchard and Jadczyk.

The integrable model of one-dimensional electrons with bond charge and Hubbard interaction is investigated at finite temperatures. The approach based on the quantum transfer matrix is employed. The specific heat and compressibility are calculated showing interesting structures at intermediate temperatures. In the low-temperature regime the Luttinger liquid picture is verified.

We introduce an iteration rule for real numbers capable to generate attractors with dragon-, snowflake-, sponge-, or Swiss-flag-like cross sections. The idea behind it is the mapping of a torus into two (or more) shrunken and twisted tori located inside the previous one. Three distinct parameters define the symmetry, the dimension, and the connectedness or disconnectedness of the fractal object. For some selected triples of parameter values, a couple of well known fractal geometries (e.g. the Cantor set, the Sierpinski gasket, or the Swiss flag) can be gained as special cases.

The quantum mechanical analogue of the classical Perk - Schultz model is considered which comprises the Uimin - Sutherland model and the integrable t - J chain. The quantum transfer matrix of these systems is established and the eigenvalue equations are obtained by an algebraic Bethe ansatz. Only the largest eigenvalue is needed for the calculation of the free energy of the quantum chain at finite temperature. The Bethe ansatz equations for the leading eigenvalue are transformed into a set of integral equations for some appropriately defined auxiliary functions. Furthermore, the eigenvalue of the quantum transfer matrix is expressed in terms of these functions. The integral formulation allows for taking the limit of infinite Trotter - Suzuki number analytically. The low-temperature limit of the free energy is obtained analytically and for intermediate temperatures numerical results are presented.

Spin generalizations of both the elliptic Calogero - Marchioro - Wolfes model and the nonlinear Schrödinger model are studied. These models are three-body problems with two- and three-body potentials, and mathematically related with the exceptional root system of type . We construct the integrable differential-difference operator, the so-called Dunkl operator, based on the infinite-dimensional representation for solutions of the variant of the classical Yang - Baxter equation. By use of these operators, we investigate the integrability and the scattering matrices.

We extend the study of diffusional relaxation on the random sequential adsorption of dimer introduced by Privman and Nielaba, to fractal media. The effect of added diffusional relaxation on the deposition of dimers in such disordered substrates makes full coverage in certain cases possible. We observe that in these cases, the limiting coverage is approached according to , where is the spectral dimension of the substrate. The jamming coverage is analysed for different fractal substrates.

Stationary states in KPZ-type growth have interesting short distance properties. We find that typically they are skewed and lack particle-hole symmetry. For example, hill-tops are typically flatter than valley-bottoms, and all odd moments of the height distribution function are non-zero. Stationary-state skewness can be turned on and off in the (1 + 1)-dimensional restricted solid-on-solid (RSOS) model. We construct the exact stationary state for its master equation in a four-dimensional parameter space. In this state steps are completely uncorrelated. Familiar models such as the Kim - Kosterlitz model lie outside this space, and their stationary states are skewed. We demonstrate using finite size scaling that the skewness diverges with systems size, but such that the skewness operator is irrelevant in (1 + 1) dimensions, with an exponent , and that the KPZ fixed point lies at zero-skewness.

We develop the formal density-functional theory of dipolar fluids allowing for bulk orientational (ferroelectric) order. The long-range character of the dipole - dipole interaction is treated by separating the direct correlation function of the fluid into short- and long-range parts. The contribution from the long-range part of the dipole - dipole interaction is shown to determine the energy of the macroscopic electric field, which depends on the sample shape and on boundary conditions. The short-range part of the direct correlation function can be used to calculate the regular contribution to the free energy, which is shape independent. An explanation is proposed for the failure of all existing density-functional theories to describe the behaviour of strongly dipolar fluids as observed in computer simulations.

We generalize the statistical mechanical theory of vulcanization to the case of D-dimensional polymerized manifolds. Starting from a continuum model of self-avoiding manifolds, we study the effects of introducing random crosslinks between monomers on (in general) different manifolds. As for the case of linear polymers, one observes a continuous phase transition from a fluid to an amorphous solid state, characterized by a finite fraction of localized monomers. We compute this fraction, as well as the typical localization length near the transition.

A partial differential approximant (PDA) analysis of series for the tricritical point of the mean-field model of Glasser, Privman, and Schulman has been developed. All features of the tricritical point are exactly reproduced by an overwhelming majority of the trial PDAs in contrast to the mixed success of the slicewise Padé approximant method. The effect of noise on the approximant convergence is also studied.

The periodic orbits of the strongly chaotic cardioid billiard are studied by introducing a binary symbolic dynamics. The corresponding partition is mapped to a topologically well ordered symbol plane. In the symbol plane the pruning front is obtained from orbits running either into or through the cusp. We show that all periodic orbits correspond to maxima of the Lagrangian and give a complete list up to code length 15. The symmetry reduction is done on the level of the symbol sequences and the periodic orbits are classified using symmetry lines. We show that there exists an infinite number of families of periodic orbits accumulating in length and that all other families of geometrically short periodic orbits eventually get pruned. All these orbits are related to finite orbits starting and ending in the cusp. We obtain an analytical estimate of the Kolmogorov - Sinai entropy and find a good agreement with the numerically calculated value and the one obtained by averaging periodic orbits. Furthermore, the statistical properties of periodic orbits are investigated.

A differential calculus is set up on a deformation of the oscillator algebra. It is uniquely determined by the requirement of invariance under a seven-dimensional quantum group. The quantum space and its associated differential calculus are also shown to be invariant under a nine generator quantum group containing the previous one.

A generalization of the classical gauge theory is presented, in which compact quantum groups play the role of the internal symmetry groups. All considerations are performed in the framework of a noncommutative-geometric formalism of locally trivial quantum principal bundles over classical smooth manifolds. Quantum counterparts of classical gauge bundles, and classical gauge transformations, are introduced and investigated. A natural differential calculus on quantum gauge bundles is constructed and analysed. Kinematical and dynamical properties of corresponding gauge theories are discussed. Particular attention is given to the purely quantum phenomena appearing in the formalism, and their physical interpretation. An example with quantum SU(2) group is considered.

We consider the solution of a class of second-order ordinary differential equations not possessing Lie point symmetries by group theoretic means. The method involves increasing the order of these equations using homogeneity symmetry. The solution of this new third-order equation is then sought in the instances that it and the new reduced second-order equation possess additional symmetries. As a result the number of second-order equations solvable by the Lie theory of extended groups is increased.

The quasiclassical approximation to the eigenvalues at the bottom of a one-dimensional potential well are obtained by applying the `twisting trick' of Simon and Davies to reduce the problem to an ordinary Rayleigh - Schrödinger problem. The key is that the twist occurs on a scale intermediate between the two scales of the problem.

If a non-autonomous quantum system has an semisimple Lie algebraic structure and its Hamiltonian can be treated as a linear function of the generators of a semisimple Lie group, we show a method for finding a set of gauge transformations that transform the Hamiltonian to a linear function of Cartan operators. The exact solutions of the equations of motion, as well as a set of time-dependent invariant operators which commute with each other, are obtained by the inverse gauge transformations. An SU(3) model serves as an illustration.

The adiabatic evolution of two doubly degenerate (Kramers) levels is considered. The general five-parameter Hamiltonian describing the system is obtained and is shown to be equivalent to one used in the Jahn - Teller system. It is shown explicitly that the resulting SU(2) non-Abelian geometric vector potential is that of the ( SO(5) symmetric) SU(2) instanton. Various forms of the potentials are discussed.

Quantum mechanics usually describes particles as being pointlike in the sense that, in principle, the uncertainty, , can be made arbitrarily small. Studies on string theory and quantum gravity motivate correction terms to the uncertainty relations which induce a finite lower bound to spatial localization. This structure is implemented into quantum mechanics through small correction terms to the canonical commutation relations. We calculate the perturbations to the energy levels of a particle which is non-pointlike in this sense in isotropic harmonic oscillators, where we find a characteristic splitting of the usually degenerate energy levels. Possible applications are outlined.

The density of zeros of the partition function of the Ising model on a class of tree-like lattices is studied. An exact closed-form expression for the pertinent critical exponents is derived by using a couple of recursion relations which have a singular behaviour near the Yang - Lee edge.

A gauged SO(3) symmetry is broken into its closed subgroups by Higgs scalars belonging to the irreducible representations characterized by j = 2, 3, 4 and 6. Explicit matrix decompositions of the irreducible representations of SO(3) in terms of the irreducible representations of the closed subgroups are made manifest. Analogous structures between the line defects of liquid crystals and the cosmic strings are notified.

We prove separation of variables for the most general (-type) periodic Toda lattice with a Lax matrix. It is achieved by finding proper normalization for the corresponding Baker - Akhiezer function. Separation of variables for all other periodic Toda lattices associated with infinite series of root systems follows by taking appropriate limits.

The concept of eigenfunction expansions for the wave equation is generalized to open systems, in which waves escape to the outside. These non-conservative systems are non-Hermitian in the usual sense. It is shown that the natural framework is an eigenfunction expansion within a two-component formalism that treats the wavefunction and its conjugate momentum together. Provided the system approaches spatial infinity rapidly `without tails', and possesses spatial discontinuities, the expansion in terms of the eigenfunctions (which are now quasinormal modes) is shown to be valid.

For a broad class of open systems described by the wave equation, the eigenfunctions (which are quasinormal modes) provide a complete basis for simultaneously expanding outgoing wavefunctions . In this paper, the linear space structure associated with this expansion is developed. Under a modified inner product, the time-evolution operator is self-adjoint, even though energy is not conserved for the system alone. Thus, the eigenfunctions are mutually orthogonal. Consequently, the usual tools of eigenfunction expansions can be transcribed to these open systems. As an example, the time-independent perturbation theory is developed in straightforward analogy with quantum mechanics, giving the shift in both the real part and the imaginary part of the eigenvalues .

A quantal guiding centre theory is presented which allows a systematical study of the separation of the different time scale behaviours of a quantum charged spinning particle moving in an external inhomogeneous magnetic field. A suitable set of operators adapting to the canonical structure of the problem and generalizing the kinematical momenta and guiding centre operators of a particle coupled to a homogeneous magnetic field is constructed. This allows us to rewrite the Pauli Hamiltonian as a power series in the magnetic length making the problem amenable to a perturbative analysis. The first two terms of the series are explicitly constructed. The effective adiabatic dynamics turns out to be in coupling with a gauge field and a scalar potential. The mechanism producing such magnetic-induced geometric-magnetism is investigated in some detail.

Previously derived expressions for moments of spectral density distribution of an N-electron Hamiltonian defined in a finite-dimensional model space spanned by a set of spin-adapted antisymmetrized products of orthonormal orbitals (full configuration interaction space) are reduced to the low electron density limit, i.e. to the case when the number of electrons is much smaller than the number of orbitals. The limit of a very large number of electrons is also considered.

Les signaux ayant des spectres de puissance dont la décroissance est en ont traditionnellement vu leur exposant estimé par une régression linéaire sur un graphique log - log. Un problème méthodologique survient lorsque le spectre de puissance du signal possède des zéros: cela introduit en effet des pôles dans la représentation log - log rendant toute régression au niveau du spectre très périlleuse. Nous présentons une méthode qui permet d'éviter ces valeurs extrêmes en ne faisant, au niveau de la représentation log - log, qu'une régression sur un sous-ensemble choisi de maxima locaux. D'autre part, il est bien connu que la dimension fractale de fonctions de Hölder d'exposant auto-affines est , ce qui permet, en particulier, de pouvoir juger de la qualité d'une estimation obtenue par la méthode précédente pour la dimension fractale de la fonction introduite par Kieswetter. Le comportement de la méthode est analysé à partir du spectre de puissance de fonctions lisses, sans que le résultat estimé ne puisse être utilisé pour obtenir la dimension fractale du signal (qui n'est pas auto-affine). Finalement, nous examinons les limitations des méthodes d'analyse spectrale pour l'estimation d'une vitesse de décroissance algébrique.

For partial differential equations written in conservative form a remarkable link between potential symmetries and direct reduction methods of order two is enlightened.

We consider the generalized chiral on with a U(1) gauge field coupled with different charges to both chiral components of a fermionic field. Using the adiabatic approximation we calculate the Berry phase and the corresponding U(1) connection and curvature for the vacuum and many particle Fock states. We show that the nonvanishing vacuum Berry phase is associated with a projective representation of the local gauge symmetry group and contributes to the effective action of the model.

We discuss some of the integrable lattices introduced recently by R Yamilov. We demonstrate that they are closely related to the usual Toda lattice by means of a sort of Bäcklund transformations. We also apply the general procedure of integrable discretization and obtain their integrable finite-difference approximations. These novel integrable discrete-time systems are also related to the discrete-time Toda lattice by means of the Bäcklund transformations. The whole construction exploits the tri-Hamiltonian structure of the Toda lattice.

New ansätze reducing nonlinear heat and wave equations to a system of ordinary differential equations are proposed. These ansätze can be constructed with the help of symmetry operators of the system corresponding to the equation under study. Exact solutions of the considered equations are obtained. The link between the Bäcklund transformations for a given equation and the conditional symmetry of the corresponding system is discussed.

The passive advection of tracers in the field of three identical point vortices is considered as the hydrodynamical analogue of the restricted three-body problem. The chaotic motion is analysed by means of two-dimensional maps, its parameter dependence, Lyapunov exponents and topological entropies. The latter can be obtained as the growth rate of a dye droplet's perimeter in time. Similarities and differences of the vortex and gravitational problem are discussed.